Most GT12 racers prefer the Schumacher Supastox diff,
the end of the diff side axle does not rotate inside the wheel so the wheel can be tightened down properly. The Mardave diff works but you have to maintain and adjust it regularly.
Technically there isn't any rubber tyres for Mardaves, and there never have been. But Mini wheels just happen to be the same diameter and width, although with ribs to locate the tyre. Make sure the rubber tyre is lined up properly on the wheel before gluing and you should have no problem getting them on straight.
Your motor will be getting hot because the overall gear ratio (how far the car travels for one revolution of the motor) will be increased greatly by changing to bigger tyres. You can compensate for this by fitting a smaller pinion gear and a bigger plastic spur gear but with touring car size tyres you will find it hard to get spur gears big enough to compensate, then you will probably not be able to get the motor far enough forwards to clear the bigger spur gear.
If there is a lot of extra load on the motor the car won't go any faster because the motor can't overcome the extra load to get up to speed.
You will get a lot more enjoyment out of your Mardave if there is a club near you where they race them, they are pretty popular cars. There is a list of clubs on Centralbooking
to see if you can find one local to you.