What types of mathematics is needed for a undergrad pursue theoretical physics? How rigorous does it have to be and how much proofs is there?
This, you get enough of the non rigorous stuff when you study physics. Of course it is easier to do courses that just do the computational parts but that would be just to make the physics courses easier rather than learning anything in itself.In my opinion, as long as you can handle it, the more rigor the better.
It depends which books you are using, most of the physics textbooks prefer physical intuitionIf you learn your math too rigorously, you'll end up being frustrated with your physics teachers and depressed by physics textbooks ;)