RC model cars in teh UK Run on 27mhz or 40mhz. If its 40mhz then yes its fine to use for Combat robots if not then it is not allowed.

I would strongly advise against plywood. It is simply not strong enough these days you would need something like 10mm polycarbonate to survive against spinners.

Drive. Those wipermotors to me would seem fine but you will need large wheels to get any sort of a decent speed with them. Another cheep option is drill motors.

Speed controlers. The 30amp electrolises sold by technobots seem fine. If you can get the ones with external relays as they are much better.

Batteries: That comes down to two things cheep and heavy or more expensive and light. If you want cheep you will need about a 5amp sla for to drive drills or ww motors. If you want to go slightly more expensive you can get 3amp Ni-mhs from technobots for around 40pounds and these will do nicely.

I hope this helps.

Regards
Ian