[MMM]

1) が never comes right after に.

2) I think they both are fine. I would probably say

私の猫が木の上にいます。

3) Yes, なに can be used by itself, but it is quite direct and literally means "What?" as in "What did you just say?"

何か means "something" (just as どこか means "somewhere", etc.)

何か by itself would only be used to say "[Is there a problem with] something?" and should probably be avoided.

In polite speech, say 何ですか.