Hi,
I think the cooper would be a good choice. The cars look great, and have a "stunt" heritage from the Italian Job movies. I think they would certainly go down well with the punters.
Apart from the midlands gearbox, they are pretty bulletproof.
Early models are now cheap. There are also lots around at wreckers now, so parts can be found cheaply.
You'd want to drive the CVT "automatic" before buying one. It certainly doesn't drive like a normal auto, ie it doesn't rev, and IMHO doesn't feel "sporty".
With regards to the wheels it is hard to go past the stock wheels for your purpose. I have 14 spare 16" rims in the garage which I have picked up for between $50-100 each. They are pretty easy to find, as people sell them when upgrading to aftermarket. The big advantage of the stock wheels is they are incredibly tough

I have done over a dozen rallies, with lots of jumps, bumping up and down kerbs etc. Also I have done loads of circuit stuff, bumping of kerbs. I have also had some high speed impacts. In all I have now done 50,000km of racing / rallying in the car, and although I've scratched plenty of rims I've never bent one

By comparison, I've seen aftermarket rims rendered useless after one rally. The other good thing about the stock rims is they have a run flat bead, which is an advantage in rallies, as it keeps the tyre firmly on the rim, and stops minor deflations on big impacts and jumps etc. I assume that would help with stunt work as well. You can probably find the stock 17" rims even easier, as they are unpopular for motorsport due to their weight.
Cheers
Robbo