It's the past subjunctive, used to refer to a wish in the present. It's related to the 2nd conditional. Here, the tone is that of a polite request.

I don't think it is either of those, although I don't know much formal grammar. The "didn't" here is short for "did not do something". e.g. "I'm going to ask Alice out" / "Id rather you didn't" = "I would rather you did not ask her out" = "I would prefer that you do not ask her out". I think that the did is an auxiliary verb for the negation of the missing/implied verb. How else would you phrase this, without 'did'?

Hey dolco, how are you? I will provide you with information on the usage of the construction ''...would rather + pronoun + verb (in the past form) +...'': 1) If you say that you ''would rather'' do something, you mean that you would prefer to do it. In speech, ''would rather'' is usually contracted to ['d rather]. If you write down what someone says, you usually write ['d rather]. E.g.: ''I'll order tea. Or perhaps you'd rather have coffee.'' E.g.: ''What was all that about?'' – ''I'm sorry, I'd rather not say.'' *NOTICE: In sentences like these you should use an infinitive without ''to'' after ''would rather.'' 2) You can also use ''would rather'' followed by a clause to say that you would prefer something to happen or be done. In the clause you use the ''past simple''. E.g.: ''Would you rather she came to see me?'' E.g.: ''May I go on?'' – ''I'd rather you didn't.'' Hope that helps! Should you have any further questions/doubts regarding English grammar, do not hesitate to contact me. I shall be pleased to help.

I'd rather you hadn't asked. I would rather/prefer that you had not asked. Seems hypothical. A situation that did not happen. . I could eat that cake for you. I'd rather you didn't. Seems also to refer to something not happening that could happen.