Mech

Ok, so it's not the conventional manual/electric shift I though it was. It's the hydraulic auto. And the PCM isn't the one I was thinking of either. My bad jumping to conclusions without checking.
So I had a read and it looks like you need to ignore the PS until the engine is running. The PS, gear shift and engine control/start though might both have a common cause. In every section I read it kept saying the same things. Check all the power supplies to all of the control units and the actuators, and check all the earths to all those same things. Then the next thing apart from the actual PCM being crook, or the dash being crook, is the communication between them.
Looking at the systems individually it should be fairly easy to check the crank position sensor, the injectors have switching power, that power is getting to the coil, and it's peak voltage, and the fuel pump is running and got pressure. For testing things that only have very momentary pulses, such as the injectors and the crank position sensor it's best to use an old fashioned analogue gauge with a needle. The needle will flicker with a momentary pulse where the multi-meter with it's slow count can very easily miss a pulse.
The transmission. The gear position and angle change switches are both suspects if all the power supplies and earths are good. The switch tests are a bit complicated but can be done with a multimeter.
But in every case it keeps saying.. power and earths. On cars and trucks when we get a whole bunch of problems, it's often all caused by a bad power or earth, and they can sometimes set trouble codes, and often they do not set codes.
The manuals mostly do tell us everything we need to diagnose problems, and in that manual it mentions the conditions under which lights and indicators flash or don't flash, and codes get set or don't get set, but without being there checking all the combinations of conditions and symptoms It's a bit beyond me to figure. If you read each section though, right through, going back several times if necessary as you start to understand the interconnections between systems, you'll probably eventually realise which lights and indicators are behaving normally in staying on or flashing, but not actually meaning there is a faulty part in that system.
Concentrate on getting the engine going, then the other systems can be tested, but keep those other systems in mind as you look at the engine problems, they are likely all connected.
Good luck with the reading and the checking. Let me know anything interesting you find and I'll have another read and hopefully we'll be able to get this thing fixed without a PCM, or, at least be confident it is the PCM before splashing out..

As Gw says, some parts are availiable, some genuine and some aftermarket. You really just need to get online and look for the part you want.
A model and market and/or having the frame number is going to be required.

I think you need to clean the rust off. In the two manuals I looked in it says the timing mark is a single small groove/line. It looks like a small chisel mark in the flywheel. There are other ignition advance marks but they are one long mark with a short mark either side.
 

If you can take the pull start cover off you will be able to turn the crank by hand or a socket, or, put it in a forward gear and gently nudge the bike forwards so it turns the engine over. If the timing mark goes right past the hole then you'll have to nudge it right round two whole turns of the crank so watch carefully and nudge it gently. If you have the spark-plug out it will turn easily, and if you have the cam cover off(if it has one) you will see when it's getting near the right place before the mark gets into the timing hole.

T for top and F for firing.

I found a BAV owners manual which I've just put into the owner manuals section.. It'll appear after it's been approved.
There's not much helpful in it but it might be handy.

If it has universal or cv joints then you will be able to see what bits turn and which don't.
If it's a solid axle (which I think it will be) then it's probably the hub the wheel attaches to that's stripped.

Go to Babbits and they have all the parts diagrams.. Look it up there and get the part number and then start looking for where it's availiable and at a good price.
https://www.babbittsonline.com/oemparts/c/yamaha_atv/parts

If it died right out, as soon as the bike moved, then I'd be suspecting an electrical problem.. 
But I'd probably try adjusting/checking the idle mixture setting first in case it is that simple.. I wouldn't pull a carb off though till I'd discounted the electrical problem as a cause. Which could be done by attaching a timing light and watching the flash as you tried to move forwards. If the flashing stopped before the engine stopped rotating, then it is an electrical problem.

Yeah the plunger needs drawing out for choke, and they only work at idle, and it needs to be a slow idle, with the butterfly or slide closed right down. They draw extra air and fuel around past the butterfly or slide and that doesn't work unless the throttle is closed off .

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.
output.ps
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.

output.pdf

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.