On the magnet thing I do like the idea of a rolling magnet idea, the magnet being round will always contact the arena floor for maximum attraction. then use normal wheels for drive. there are lots of problems with this idea like how to turn the robot or what happens if the magnet picks up some steel like loose nuts or washers. but there are ways to mitigate this.

The pushing power is down to the friction between the tyres and the arena floor. by increasing the attraction between the floor and the tyres the pushing power will increase, but also the drive transmission load will be greater!

I say give it a go! Invent something new and refine the idea.