2011 King Quad 750 tires
 

I'm looking to upgrade from my stock tires. I've pretty much decided I want the Kenda Bear Claws. My question is: If I go with the 26x11 12 for the rear an the 26x9 12 for the fronts, can I still use the stock wheels?

Yes you can

After further research, I've decided I like the bear claw htr tires better. I would like some opinions on whether to stay at the stock 25 inch size or bump up to the 26 inchers.

Good choice! The HTR's are awesome tires. Personally, I would stick with the 25 inch tires. They will still be significantly taller than the stockers anyway, and will minimize the power loss and higher gearing.

All that matters is the last number (12) 12 I.Dia. tires go on any 12" rim.

Its a 750, power loss will be almost unnoticeable in your case.

Would the only real advantage of the 26 inch tires be 1 more inch of ground clearance? Also, would I need a lift for 26 inch tires?

The 26" tires will only give you another 1/2" of ground clearance (half the height is above the center line of the wheel). You won't need a lift either. However, the 750 already has higher gearing, and the increased tire size will raise the gearing even more. Contrary to what other might think, you will notice some power loss (do to the lost gearing and added weight) even with your 750.