"I hope not" is the standard response. Nobody says "I don't hope so". In theory, it is grammatically correct ( we can say "I don't think so", for example), but it is never used. In terms of usage, therefore, "I hope not" is the only correct response.

A and B are all correct but A is appropriate because it makes your speaking have a good flow.

In English we say A) I hope not - as in, I hope (it will) not (rain). You could say (B), but it is more common to say (A).

"I hope not" is more direct. You are directly answering the question with a negative response because you are hoping it does NOT rain. Either are grammatically correct in this case.