Title is self-explanatory. I wanted to major in math but I can’t write rigorous mathematical proofs. I can’t major in applied math at my school.
Depends on the program.
I might be wrong, but I think most statistics programs require at least one or two mathematical statistics classes, so it is possible that you will have to write a few proofs.
You’ll probably have to do some proof work but it’s mostly to understand the theory, not to use it on a regular basis. The proofs also aren’t as terrible as the ones you do in set theory - they’re proofs of the probability theorems and equations so they’re more interesting and help you understand the equations you’re using.