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.

I agree Brakes and bearings would be good. Usually the brakes are packed so tight with dried mud and stuff they will hardly move, yours is moving good. It was hard to tell exactly where the play was coming from on the video it actually appeared that the whole axle was shaking, so is your play from left to right or up and down.

Yup I'd definitely be suspecting the brushes in the starter. They wear out quite regularly, and sometimes get stuck in their holder. The easy way to test them is to use two jumper leads and jump from the battery to earht, and from the battery positive straight onto the starter motor's live terminal. That eliminates the battery cables, bad connections, relays and switches. If it doesn't spin over then whip the starter out and dismantle it. The new brushes get sold with new springs and brushes all mounted on a new steel plate you just slip straight in.. Simple job. Cheap too.

Maybe, im guessing, i see more freshly rebuilt carbs running lean than rich, and usually you can smell it at the exhaust if its rich enough to bog, plus smoke at the exhaust.

I think the mixture at idle is too rich. That would explain it idling cold but then cutting out once it warmed up. And winding the idle mixture screw in doesn't make it too lean by the sound of it... Both things point to a rich mixture. Check the air filter is clean and oiled, and not over-oiled. Check the breather hoses on the tank and carb are fitted correctly and not blocked. The dieing when warm could also be a tight valve clearance so it would be a good idea to check those. The idle mixture needs to be adjusted at the slowest idle speed it will idle steadily at. So you slow it with the idle speed screw then wind the mixture screw in and out till it is in the place that gives the highest idle speed, then you wind the speed screw out to slow it right down and you adjust the mixture to it's best idle again, then slow it, then adjust it, and you keep going around in a circle slowing it and adjusting the mixture. Once it's idling slow and steady then you can speed it up if you want to but leave the mixture where it's best at the place where it will run nicely with the least amount of butterfly opening. If it doesn't come right, if the idle can't be adjusted in both directions with similar rev changes, and/or it keeps dieing once warm then I'd check the float level, which can probably be checked with the carb in place using a bit of clear hose, and I'd check the jetting.

Gw's right about in being leaner, but try turning the mixture screw in and out making sure there is a sweet spot in the middle, and that winding it too far either direction makes it run bad.. That checks the float level's not too high or low, that there are no bad air leaks, and that all the jetting is about right. Checking it's responsive in both direction often reveals a problem. And yeah, stale fuel is something to watch for.. We generally do suggest stale fuel if it sounds like the bikes been parked up for a while.. Good work changing that...

Turn it in to make it leaner.

My first thought is it might be running too lean by mixture setting or could possibly have an intake manifold leak. Im assuming oil is good and filter changed and getting good oil flow. When it shuts down and its hot will it start back up, Also make sure all your cooling fins are not caked with debris. I dont remember if that model had a cooling fan i dont think it did, but if its there check for proper operation.