Mech

Ok. If it doesn't read 2Wd, 4Wd, Diff-lock on the shift lever then it doesn't have a diff lock and is just a limited slip diff. The test for their functioning is as described earlier, jack up one front wheel and try to turn the other. It should take some specified amount of torque to turn the lifted wheel.. It's stated in the manual.
And apart from being larger diameter wheels, if the new tyres have a flat tread/foot-print, then yup, they will really play havoc with the steering, and handling.

This is what my book shows for the 2000-2002, but the older models in the book are the same.. Just two wires.
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The book shows a four wire switch up to 1995...

I doubt the flare nut type would be made for that torque.. 
To tell the truth though, I doubt it needs to be torqued precisely. If you have a torque wrench that goes to 150, or close to it, put it's drive in the vice and get a feel for what various torques feel like, like start at eighty and then try one-hundred then one-twenty or whatever your wrench will go to, then just make an estimate and do the inner nut up reasonably tight and then crank the outer/lock nut up to as close as you can calculate it should be with a plumbers wrench/pipe wrench or big crescent.
 

The kill switch wires are just a parallel connection to the main/key switch's wires, branching off the same black and brown wires the main switch use, and since the main switch works, it has to be in the branching wires that the problem is.

Savage is normally onto it but I don't think the start circuit is any problem.. He might elaborate though.
I'd unplug the main switch and use the buzzer on the gauge to check that it all switches as it should.. I'm suspecting there's some wear stopping the sliding contacts moving as far as they should, and possibly only not as far as they should in one direction, or dirty contacts on the kill part of it..The sliding bit might be contacting the power ok but not a dirty kill contact, or there might be two sliding contacts and one is dirty of loosing travel.. You'll have to try and figure it with the gauge, unless it comes apart in which case rip into it.

Put tyre slime in them...  It fixes slow leaks and prevents new ones.
If it is the beads leaking you'll need to pull the wheels and lay them on the side to get the slime there, but if it's prickles in the tryes just riding it will seal them..  I love the stuff. I put it in as a preventative.
Haha.. I'm a mech Gw.. we never do anything to our machines unless we have to..

The lights are led, and the rest of the display is LCD.. Are the lights dim as well, or just the gauges etc ? Lcd needs a good power supply.. You could check the power supply and earths.
It's also possible the dash is meant to dim if the lights are on..  I'm not sure about that, but it's a possibility. The owners manual might mention it. The service manual I looked in didn't mention it. You might try turning the lights on to see if it makes a difference.. It shouldn't, because they are already dim.. haha.. but we never know what might reveal itself. Turning the lights on will turn on the dash lighting and put more load on the earth and/or earth.. I think..
 

No Ps, those are for welding tips and they are like tiny files. Don't use those.
If jets are dirty it's nearly always a lump of dirt or bit of fluff and you can see it if you hold the jet to the light. Most things can be blown out easily. Sometimes the jets get a build up of verdigris which doesn't blow out, and people use chemicals, but I just use any small bit of hardwood whittled down to a fine point to ream then out, and then give them a blow through and reinspect against the light. It's important to take out the emulsion tube, which is the brass tube the slide needle drops into. A few taps from the top with the handle of a small screwdriver and it comes out the bottom. It has several tiny holes halfway up the tube that need a good blow out because they are too small even for a whittled down bit of wood. Don't try cleaning them with a pin, a single wire from a wire brush some times fits nicely but they mustn't be enlarged or distorted.. or blocked with a tiny point of whittled down wood !!  The holes in the tube are arranged on opposite sides of the tube, so you can look right through if you are very careful, and you will see light if they are all clear. The drilling in the carb body that the emulsion tube fits into, is a precise diameter and volume and needs to be cleaned out of it has a build up of chalky or gummy crud..
Apart from the fuel jets there are air jets near the air-cleaner end and some of those can not be removed, and even the ones that can be removed don't really need to be taken out. They are so big you can see and blow straight through them. Blow every passage you can in the carb body, in the opposite direction to normal fuel or air flow, as much as possible. Blow a little one way to see where it comes out before blasting it through from the other way. The idle discharge holes are the smallest passages in the carb body.. Try not to force any fluff or dirt up towards them by blowing from below. It's awkward, but best to open the butterfly or take the slide out, and blow down through the idle discharge holes as best you can, before blowing back the other way from below.
If you have compressed air it's a good thing to make a long tipped air nozzle out of four mill steel brake pipe. It's handy for getting down the hole where the slide drops in, and blowing the drillings there or under the butterfly.
 

You were holding the revs up too long...  Keep gently cycling it always having the motor either accelerating lightly or decelerating sharply..  And use less throttle..  I bet I could fill your yard with smoke out of that thing..
I think you will find the oil rings are not doing their job.. Clean the breathers first though.. make sure you can blow through all the hoses except for the one with a restriction.. that should be harder to blow through but it should flow a small amount consistently. If you find a blocked breather, after clearing it, try riding it so it has long runs of steady, heavy load, without any sudden or hard acceleration and only gentle deceleration. Find a hill, use higher gears wherever possible, and don't let it rev away under light loads, change up as soon as possible..  Sometimes with a blocked breather the oil that gets blown up gets under the rings and they start floating on it.. If we can put a steady compression on it more than not, the rings slowly scrape the oil away and the rings reseat metal to metal.
You might get lucky.

Yup, I'd try a different tool, possibly a torx if one felt like it might tap in reasonably tight.
Next would be as suggested, try to tap it around, You need a small chisel that's not too sharp, so it will dig in if hit straight down, but won't cut metal away when we lower the angle to get the bolt to move. If not  a chisel, then an engineers punch that is ground off nice and square so it it sharp and will dig in.-Then a couple of sharp hits downwards to get the chisel or punch to bite in, then one light tap to try and move that side around, then go to the opposite side and repeat, two sharp hits to get the punch to bite, then lower the angle and give a tap or two, then go back to side one and give that a tap, then side two...  and so on.
If neither of those works then drill it out in stages, trying to loosen it by taking the center out first, and trying plan two above, then drilling it bigger and trying an easyout, then drilling it right out and re-tapping if necessary.

I couldn't see the link Params.
The coil will only make one spark every time you disconnect the power from it's two primary terminals. The spark will be from the HT lead to the negative coil terminal.
If you connect the sparkplug and the jumper to the coils negative terminal, and then just rub the positive jumper clamp on the primary positive terminal it should make and break contact as it scrapes along and make several sparks.

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Look the parts up on an online parts place like partzilla and they will likely have some way or where, that they will tell you what else that parts fits.
https://www.partzilla.com/product/yamaha/5GH-17660-00-00?ref=a5a2e51fd1b12c786666c251d29374489d129825
At the top of that listing click on the button called "clutch" and it will take you back to a parts diagram where you can check each part..

https://partshawk.com/wix-wl10090-engine-oil-filter.html

I'd check the gauges earth wire is ok..  And perhaps have a look into the tank to see if the float isn't somehow jammed upwards.. 
If you download the service manual  it will tell you exactly what to check and lead you through a step by step process to diagnose it.

Onya..  well done.
 

To adjust the slide needle you take the top cover off, pull the slide up, take the needle out taking careful note of where the spacers above and below the C clip go(look carefully, they are different thickness), Then shift the C clip up or down to another groove in the needle. Putting the clip higher makes it leaner.

I don't have one or know much about any of them in particular, but I know that you can do quite a bit of diagnostics using the service manual and the dashboard display on most of them.
The engine anyway is fairly well supported.. the EPS maybe not so much. 

My internet's too sh** to see the video today Parham, but good work getting it going.. I'll try looking later.
I don't know what technique you use with the pull start, but the suzuki owners manual suggests an unusual way of using  the decompression lever when using it. Most things you just lift the decomp and then pull the starter. Suzuki say to lift the decomp and then pull the starter very slowly till the decomp lever just flips itself back to the closed valve position, then let the rope right back in, gently engage the pull start and then give it a hard pull. It's a good system. It means you've parked the engine with the exhaust valve just closed and all ready to start an intake stroke, and still leave the motor enough turns to get up to speed before the next compression.. You get full charge of air and fuel, easy pull and a reliable start..
Ha.. and now my internet goes and I see you follow suzuki's way already.. 
I only read about it a while back after years of struggling to get bikes to pull start.. 

Put it in two wheel drive, high ratio, and try rotating the rear wheels..  While you have it up.

Hi. Yes that's a good explanation. I'd suspect that's in the steering linkages then.
There are two rods that come from arms on the wheels into the center of the bike and attach to a short lever that pushes one of the rods as it pulls the other rod when you turn. We can adjust those rods to change the toe in of the wheels. When the handle bars are straight ahead those rods should both be the same length and the wheels should(on most four wheel drive bikes), be slightly wider apart at the front of the wheels than at the back of the wheels. That's known as toe out.
Now, when we turn a corner, the wheel on the inside of the turn needs to turn a little more than the outer wheel because the inner wheels is following a smaller radius turn. We arrange that by having short levers on the steering hubs that don't come straight back from the hubs but converge slightly towards the back of the bike. The short levers on the steering hubs point to a point on the rear diff. What that does is, as one rod pushes it's steering arm, say the left wheel arm, it swings it's lever through the point where the rod and lever are at right angles. The rod starts off pushing the arm at slightly one side of a right angle, through right angle, and beyond till the rod is past a right angle with the arm. That wheel turns at the optimum amount possible for the rod travel. The other side rod though starts off pulling it's arm already past a right angle, and keeps on pulling it more and more into a position of decreased movement in comparison with the rods travel. The left side steering rod achieves full swing of it's wheel, while the right side rod struggles to turn it's wheel much. The left wheel tucks into a tight turn, but the right wheel doesn't turn as much as it might and it follows a bigger radius turn.  All this is called "toe in on turns", and it's achieved by having the short arms attached to the steering hubs converging. The whole idea is known as the Ackermann principle.
I'm pretty sure that your bikes are both going to have something bent, probably the short steering arms attached to the wheel hubs, or, the upper and lower suspension arms have been changed for ones of different length, or, the steering shaft that comes up to the bars is twisted. People have adjusted the rods so the basic toe in when it's straight ahead is still right, but there is no toe in on turns, or at least, the toe in on turns is wrong when turning left.
On some bikes the rods attach in the center to a plate that's wide and both rods attach the same distance from the steering shaft. On those bikes both rods attach from either above or below. On other bikes the rods attach to a narrow plate attached to the steering shaft which is attached to the bars, and because the plate is small they attach one rod behind the other. On those bikes one rod gets swung through a longer arc than the other, which can also give toe in on turns. They compensate so as to not get excessive toe in on turns by attaching one rod from above and one rod from below. That works because the steering arms out on the wheels are at a different height to the centre plate, and one rod ends up at the same height as the plate and steering arm, and if it's attached so as to swing through the shorter arc, that is it's attached to the hole in the plate nearest the steering shaft it pushes straight and gets full travel, while the other rod being attached from below, and to the further from the steering shaft position on the plate, is pushing sideways but also upwards or downwards and loses some of it's travel. That lost travel compensates for the extra distance from the steering shaft.
So, a long and complicated explanation, but in there somewhere is the answer to your problem. One wheel is getting turned more than the other wheel during turns. You need to check how those rods attach in the center, and then check for bent steering arms on the hubs, or wrong length suspension arms. I hope all my terminology is clear, but if you're not sure which bits I'm referring to at any point ask and I'll find a picture or something.
 
 

The wiring plug and pins/terminals should be availiable though from an electrical supply place. All vehicles use standard design plugs and terminals. The various designs have names.. If you identify the design of plastic plug, then you will find new terminals for it easily enough I would think. The old pins will come out of the plastic once you have released the small locking clip inside it. 
To see various designs of plug and pin you could look in RS Online... Visually identify the plastic plug and then look at what terminals fit it.. 
Industrial Solutions | RS formerly known as RS Components (rs-online.com) Electronics components, power and connectors...
Or post a picture in the forum and I or someone may recognise it..

You could fit the master, then put your finger over it's outlet while you pump the brake lever, using your finger as a one way valve to get fluid in and most of the air out. Then connect the pipes and bleed it at the back wheel in the usual way.

The workshop manual will tell you exactly where and which way each gear should be on it's shaft, with spacers etc, and which shift fork goes where.
The cases don't have a gasket, just sealer, and I'd highly recommend you buy a tube of loctite master gasket and use.. It's clean to use and doesn't leave long strands of squeezed out silicon inside the engine to block the filters..  It doesn't really matter what gear it's in as you assemble the cases(if that was the question).
As for the gear shift, I'd put the center cases together temporarily and try every gear change by turning the shift barrel from in the side case, whilst turning the two gear shafts. It would be good to have the lever and wheel in the side case that rubs on the shift barrel and holds it in each gear. That wheel both holds it in a gear once selected, but also makes sure the shift travels all the way into gear if you don't move the shift lever full travel. The shift barrel may be a little tight to turn if it's all dry in there, but you should be able to turn it with your fingers as long as you turn the shafts enough. Sometimes you will need to turn the shafts quite a few turns to allow the cogs to line up and engage, but they should all slip nicely from one gear to the next with enough turning of one or both shafts. Be patient, test it carefully up and down through all the gears.
If you can get all the gears at that point, it suggests the problem is in the outer/side case, something in the shift mechanism before it gets to the shift barrel. Possibly the return spring on the lever, or the spring on the ratchet mechanism, or the ratchet mechanism itself.
Post photos if it's easier than trying to describe things, or, get the manual or look things up in an online parts place, and describe parts as the book does.
 

You need the battery in there to do any meaningful output tests. The regulator will get hot if it's trying to control an open circuit voltage. They work by shorting the excessive power to earth, but they are also designed in anticipation of about one to two amps(make and model) load always going into the battery.
The starters wear out brushes regularly. it's an easy fix. They sell new brushes, springs etc all on  new brush holder plate. cheap !