Mech

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 !

It sounds like an electronic problem.. If you have a timing light, put that on and watch it flash as the engine stops. If the spark dies before the revs do, then the cdi or ignition coil is failing. If it starts to die out but the spark keeps flashing the timing light right down to the last couple or turns of the motor, then it won't be the ignition.

A real common problem with those is the starter brushes. They wear out and they get stuck sometimes/often.
The starter relay has two windings to operate the plunger that closes the contacts. One set of windings earths internally in the relay, and the other set of windings earth through the starter. If the starter brushes are not making good contact the starter relay doesn't get enough power through both sets of windings to pull the plunger and close the contacts. Then you get a click.
I'd use a jumper lead straight to the starter's terminal, or, now you have the body off, straight to the starter lead going to the starter. Put another jumper to an earth and the starter motor should spin. If it doesn't spin, or if it struggles to turn, then it needs pulling out and stripping and a new set of brushes. Easy job. The new brushes come with a new mounting plate and springs and all ready to slip in.