Sometimes English puts a verb in the subjunctive voice to express that the action is "viewed emotionally (as with doubt or desire)" [Merriam-Webster on "subjunctive"].
And sometimes it does not.
For example:
Why does "hope" require "do" here, while "wish" requires "did"? Apparently, sometimes, we express an emotional attitude (toward making noise) by the use of subjunctive after the word "wish," but with the plain old indicative after the word "hope."
Part of the reason has to do with the "counter-factual" meaning of second statement. "I wish you did not do that"... but you do. That counter-factual quality is not clear in the first statement. "I hope you do not do that"... and perhaps tomorrow you will not.
And maybe the reason has to do with ambiguity. (1) "I hope you did not do that every day" might be indicative: I hope (now) that you did not (in the past) do that every day. So we avoid the subjunctive there, where it might be mistaken for an indicative statement with a different meaning. But (2) "I wish you did not do that every day," doesn't seem to pose that danger; we might say (3) "I wish you had not done that every day," but we would not express that sentiment with statement (2).
Anyhow, if I'm right, and if you want to sound like other careful users of the English language, then there seems to be a case-wise need to find careful writers using the sort of construction you're considering.
“[It] is no wonder that the incident you have related should make you
nervous, and I would rather you did not sleep alone: promise me to go
to the nursery.”
–Jane Eyre, by Charlotte Bronte
“But I would rather you did not leave Paris until I have the signed
agreement in hand.”
–Life of Mozart:, Volume 2, by Otto Jahn, apparently a 2013 English translation of a letter written in German in 1778.
“May I guess?"
"I had rather you did not.”
–The Duke’s Children – Killancodlem, by Anthony Trollope
"I would rather you did not insist upon explanation so soon, but if you
do, I will try to satisfy you…”
–Looking Backward: 2000-1887, by Edward Bellamy
In those examples, the "did not" construction refers to hopes for the future. I did not yet find examples going the other way, involving "do not." But as always, we can form our separate opinions about whether the citations are old-fashioned constructions that modern authors do not emulate.
“I came to see you, and to meet your parents.”
I blushed so hot I felt feverish.
“I would rather you did not,” I said softly.
“Why not?”
“I’m only seventeen. I don’t – I’m not thinking of such things yet.”
–Girl with a Pearl Earring: A Novel, by Tracy Chevalier (2001)
That one's a recent author, but (1) it's not clear whether the "rather" business expresses a regret or a hope; and (2) you could (as always) ask whether the construction gives an archaic flavor.
To me, "I would rather you did not" sounds better regardless of whether we're considering action in the past or in the future; but I can imagine that someone would use "do not" because the use of "did not" seems ambiguous. If I say "Yesterday you parked in my driveway. I would rather you did not," do I mean that I resent what you did yesterday ("I wish you had not"), or that I hope you will act differently in the future ("I hope you do not")? Perhaps that possible misunderstanding would lead to reserving "rather you did not" to regrets about the past and "rather you do not" to hopes about the future.
Best Answer
"What would you rather I do" is correct in the case where you have presented some options (or are just asking 'what to do' more generally) but are yet to take action.
"What would you rather I did [or 'had done']" would be used where you did something already, but then maybe it's being questioned so you are asking the person -- well, what should I have done instead? What would you have preferred that I did?
"What would you rather?" does get used occasionally but I think it's a bit ambiguous. If there are several options I would rephrase that to "[what/which one] would you prefer?"