Mech

Yeah ok. So when you get the new rings, check the holes in the back of the oil groove are clear, check the side clearance of the rings (that's the clearance top or bottom of the ring), and check the ring gap of course. When you get the bike together, check it all over before you start it, then when you do start it, check the oil is getting to the head, then start riding straight away, don't let it sit idling, and when you are riding it, put a heavy load on it until it's built up about 3/4 revs, then let the throttle right off till it's slowed to just above idle revs then give it about 3/4 throttle again and keep it there till about 3/4 revs.. repeat that cycle about ten minutes or so and the rings will bed in.. That first few minutes is more important than hundreds of miles of careful running in.If you bed the rings like that, the engine doesn't need any running in. Let it sit idling and the rings might never bed in.
The rings have to have a heavy load on them in the first few minutes.. That means letting the engine labour slightly..

You do need to remove the shift motor and gears according to the manual I'm looking in. They describe it as the "ESP" models (electric shift program), and that's what yours is.
Good work getting into it though given your difficulties. Well done.

You can test them while you're holding them in your hand by blowing into the fuel fitting while you slowly tilt the carb in a way/direction that will make the float slowly swing from it's hinge at the top, towards closing the needle onto it's seat. You have to hold it so the float's hinge is going to be at the top. It should blow through freely when the floats hanging away and block off entirely when the float is hanging even just lightly against the needle.

Hi. Well the choke works by sucking extra fuel up out of the float bowl when you pull the choke plunger out, either with a cable or a knob. It draws fuel up and mixes it with air then discharges it through a drilling that comes out between the butterfly and the engine. The fuel will only be drawn up though if the butterfly is fairly well closed. If you open the throttle/butterfly while cranking the cold engine the vacuum decreases and the fuel doesn't get lifted enough to get discharged into the engine. So it's important to have the throttle adjusted well closed off as far as possible, and not to use it till the engine has had a chance to be richened up by a few cranks.
That said.. Did the choke use to work before the engine work ? Are you sure the choke is the problem ? Could it have an air leak into the sump, either a seal or a gasket somewhere, or a leak at the manifold ? One indication of an air leak would be it not idling well, or not being able to adjust the idle mixture rich enough.

Click on "recoil starter...
https://www.babbittsonline.com/oemparts/l/kaw/500ae90af8700209983b7610/1999-lakota300-kef300-a5-parts

Have you got compression ? Try poking your finger in the sparkplug hole and give it a kick. It should blow your finger out. If you can hold your finger in there then it's lacking compression.
Has it got spark ? Even though it might have spark at the plug when you kick it outside the engine, it may not spark inside the engine under compression. You could try a new plug which is a simple and cheap check.
When you try to start it, if you pull the plug out, is it wet with fuel ? If it's wet then it could be that the spark or compression is faulty, or the carb is flooding. If it's wet, turn the fuel off and keep trying for a while. If it's dry then check the fuel is getting right into the carb. Undo the drain screw on the bottom of the carb and watch as it runs out of fuel, then check the fuel keeps coming through fast enough to keep the engine running.
Answer the questions(all of them) and I'll see what I can suggest next.
And if you tell me what model it is I'll see if I can find a manual and have a read..  It may have an air valve on the carb or something I should know about..  haha.

Yup, check the fuel flow from the tank, then at the drain screw on the carb. If that's ok then take the carb out and strip it entirely, clean it carefully, and it will probably come right. I doubt it is going to be spark, and if it is it's going to be tricky to diagnose. The fuel system is easier to work on, but you have to be thorough and when the carb goes back on you have to know it's right/clean. Otherwise, if you don't know it's all real clean, and been blown through every jet and drilling, you are going to be back to guessing whether it's fuel or electric..  Make certain the fuel system is operating first.

There may not be anything wrong with the bike.
It's common for gears to jamb hard against one other if the engine stalls, and it makes them hard to shift. It can happen in forwards or reverse. First thing to try is rocking the bike backwards and forwards quite forcefully while you are trying to move the shift lever. The rocking removes the load off the gears and hopefully they will slide apart.
Another thing that often fixes it is, most quads have a feature where if you apply the brakes hard enough for the brake light to come on, then the electric start operates even though it's in gear. If you can use that feature to start it in reverse you will probably find that after you've backed up a foot or two that the gears will shift normally.
I'd try those two things first, then as Gw says, check the gearshift adjustment, and any safety interlock that's to prevent it engaging a forward gear while it's in reverse. Not all bikes have such a feature but if it has a lever or knob you have to operate to get reverse, check it's adjusted right.

It's all internal to the coil. The magnets induce a voltage in the primary windings, the voltage is allowed to flow through the windings as a current, in the old days through a set of points, now days through a transistor, then, back in the day, when the points opened and the current stopped flowing the voltage flowed into a capacitor to help prevent the points burning, but also to increase the counter flow of voltage that developed in the primary windings as the induced magnetic field collapsed. Now days a transistor stops conducting instead of points opening, and there may or may not be a capacitor in there(I don't know). The collapsing magnetic field, combined with the voltage getting reversed as it goes back out of the capacitor, causes a high voltage to be induced in the secondary windings and so a spark. The wire coming out is only a kill wire..
Some of your old bikes will have a long core with a winding that spans two of four magnets in the flywheel, and a set of points. That winding either powers a separate ignition coil up on the frame somewhere, or in old machines it powers a secondary winding that's wound right over the top of the first(primary) winding I described. The second way with the secondary wound over the primary right there inside the flywheel is mostly on stationary engines and the separate coil is more common on newer bikes, like the sixtes and seventies bikes. Real old bikes, with a villiers engine say, they used the old system too.

The three wires the same colour can be put in any order into the plug.

I think you should find your frame number and confirm which it is, the model up to 2001, or the model from 2001 onwards... Then try to find the shaft you know will fit.

I think the starter motors are fairly common and fit other makes as well. I just take the old plate with brushes to my local bike shop and the girl whips out three boxes of starter brush sets and gives me the ones that match mine. I think there are one or two makes and models of starter that do a lot of make quads.
If you get the numbers off the motor though you could search, or look up the genuine starter or parts and search for those numbers. I'm pretty sure you'll find they are common.

Coil primary 0.5Ohm, secondary 10-16K Ohm.
Pickup/trigger coil 90-140 Ohm, cdi charge coil 106-160 Ohm.

The most common reason for that oil in the airbox, and especially if it starts doing it suddenly, is that the sump is full of fuel. A sump full of fuel would also cause the engine dieng once it starts to get hot. I'd suggest checking the oil level as the first thing.
A lot of bikes also have restrictor valves or orifices in their breather hoses, and in some models if that restrictor blocks up it can cause problems like the oil in the airbox.

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.