Both are perfectly acceptable, but as another teacher mentioned, there is a slight difference in tone. The first one might come off as too direct and make the other person feel awkward. The second one is a bit more polite and not as demanding.

I'd say only the second is grammatical. I've never heard the first one. The closest correct alternative I can think of is "I'd prefer you not to ask questions".

Yes, you can use them interchangeably without much difference in meaning at all. Both are grammatically correct.

Technically yes, they do mean the same thing, but tone is very important. By using 'not ask' it is less grammatically correct but still accepted by natives, we use it often. It's very straight to the point and can be viewed as a firm or final tone. On the other hand 'didn't ask' is a little more tame and polite and is the more grammatically correct version.