Instead of "Would you mind if I ask you something?" Why past tense, I don't get it?
Why is it "I asked" rather than "I ask"?
This is simply because of tense agreement between the two clauses. Both clauses are in an in irrealis mood based on the past tense subjunctive. "I asked" is in the past tense because "Would you" is in the past tense.
Why are they both in the past tense?
This is simply because this is one way in English to express conditionals about the future. The viewpoint is that the speaker imagines being transported into an imaginary future, and then speaking about hypothetical events from that point of view as if they were already in the past. The use of irrealis constructs like "would" and "were" makes it clear that those are events that are not known to have happened.
The other way is to use a simple indicative future (and note the agreement between the two clauses again):
Will you mind if I ask you a question?
This uses simple indicative future tense. Although the second clause is in the grammatical present tense, the present tense serves as future in English, in some circumstances. "I ask" refers to a future time, and "will you" to a time beyond that one.
Here are examples of agreement in a few other tenses:
Did you mind that I asked you something? [Indicative: these situations have already happened.]
Do you mind that I'm asking you something? [Present: going on now.]
Would you have minded if I had asked you something? [Irrealis in the past.]
What is the difference between "will you mind if I ask" and "would you mind if I asked"?
The factual meaning is exactly the same: the questions ask for exactly the same information about the same hypothetical event. There is a difference in the temporal perspective, and also a difference in nuance.
In the first form, the speaker imagines that he or she has transported to a future imaginary point in time. At that time, she is asking a question and wants to know whether the other person will mind that in the immediate future relative to that point. In the second form, the speaker imagines that he or she has transported to a future imaginary point in time, at which she has already asked a question (the asking is in the past relative to that point). The speaker wants to know whether the other person minds that the question was asked.
The nuance difference is more important. The word "would" carries a softer, more polite tone. It is used in polite suggestions and requests.
What to make of the mixed forms then?
? Would you mind if I ask you a question?
This can be regarded as a derivative of "Will you mind if I ask you a question", where the word "will" has been replaced by "would" for the nuance, but without a corresponding adjustment in the other clause. It can also be regarded as a combination of the plain future with a subjunctive first clause: "If I ask you a question (in the very near future), would you mind that situation?"
I put a question mark on this because it sounds slightly inferior to "Would you mind if I asked ...", which is spotlessly clean.
* Will you mind if I asked you a question?
This combination is ungrammatical; you don't hear this. There is a glaring lack of tense agreement between the clauses.
A final note. English speakers nowadays tend forgo the subjunctive mood in conditionals, particularly in informal writing or casual speech. Instead of "if I were", you often hear "if I was". In the case of "if I asked", we don't actually know whether it is the subjunctive or indicative because the form is identical for that verb.
This is a speculative type of a conditional construction, mode, or mood.
The conditional mood is a grammatical mood used to express a proposition whose validity is dependent on some condition, possibly counterfactual. (Source)
English has three types of conditional sentences, which may be described as factual ("When I feel well, I sing"), predictive ("If I feel well, I will sing"), and speculative ("If I felt well, I would sing" or "If I had felt well, I would have sung"). As in many other languages, it is only the speculative type that causes the conditional mood to be used. (Source)
In English, when a question like this is asked, the verb associated with “would” (or “could” etc.) takes its base form (“mind”), and the verb in the conditional clause takes on the past subjunctive (“asked”).
For “you” to “mind”, we have to imagine a world where I've “asked”.
That is the grammatical usage particularly, in the conditional sentence type II ( If+past form, modal auxiliaries in the past.)