Both will be understood, and are commonly spoken, but the first is grammatically wrong. "Rather" is correctly used with an infinitive, similar to subjunctive, like this: "I'd rather swim" "I'd rather not ask questions" To say "I'd rather swim" with "you" as an implied subject, you would say: "I'd rather you swam." Your first sentence violates this natural structure by forcing "I" and "you" to compete for being the subject. This can be fixed by saying "I'd rather THAT you not ask questions" which works because "that you not ask questions" is a good clause in subjunctive mood. If you wanted to state the opposite, you would say "I'd rather you asked questions" (in which "asked" is subjunctive). Your second sentence is fine because "didn't" looks like subjunctive. Past tense can look like subjunctive for all verbs except "to be" which has the special form "were".

"I'd rather you didn't ask questions." More correct and common. Sounds polite. Good for speaking and writing. "I'd rather you not ask questions." Also okay, especially in American English. This is more direct and sounds slightly more assertive or formal. Not used as much in textbooks.

I'm a native US speaker. I believe they are interchangeable. The second one sounds more natural, more "idiomatic" to my ear. It's the phrasing I would use myself. I'm not an expert, but the first one sounds fine to me, and grammatically correct. As a matter of fact, to me it sounds British. With regard to grammar, I think this is grammatically correct: "I'd like you to call Lewis today." "I'd rather not." And therefore I think "I'd rather not call Lewis today" is correct. And therefore I think "I'd rather you not call Lewis today" is correct.

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".