Q:

Is “if they would do something” correct English?

We were discussing if the following sentence would be proper English:

I asked them if they would do me a favour.  (A)

The meaning is that I am referring to a point in the past where I asked another person (group of persons, whatever) to be so nice and do me a favour of some nature.
Somebody claimed you mustn't use "if" with "would" in this fashion, but instead say:

I asked them if they do me a favour.  (B)
  I asked them if they did me a favour.  (C)

But I think this has a different meaning, both versions (one with "do" and one with "did").
I can think of yet another option:

I asked them to do me a favour. (D)

but I suspect this has a slightly different meaning, not including the tone of the original question.

Could you bring some clarity to the matter (for the layman, please), which is correct and if the meanings of A through D correspond to the intended meaning I described (if they are correct, as the semantics of them would be undefined otherwise -- obviously).

A:

Your options (A) and (D) are correct, while (B) and (C) are incorrect for the meaning you're trying to convey.

I asked them if they would do me a favor.

The word "would" here is correct idiomatic English. If you extract the if-clause into its own sentence you get "They will do me a favor." However, when used as a subordinate clause with if, the verb will must become would in order to agree with the past tense asked.

I asked them if they do me a favor.

This is bizarre and ungrammatical. The tenses in both clauses should agree in most situations, and in any case no English speaker would ever say this.

I asked them if they did me a favor.

This is grammatically correct, but makes little sense. The tenses between the clauses agree as they should, but having both clauses in the past tense means that you're asking if they have already done a favor for you. This is not what you want to say, I'm pretty sure.

I asked them to do me a favor.

This is semantically equivalent to the first option, but of course the syntax is completely different. Since infinitives have no tense, to do is correct here. You can use this option if you're nervous about tense agreement and modal verbs.

A:

I asked them if they would do me a favour. (A)

Sounds just fine. It's an indirect quote of "Would you do me a favor?"

I asked them if they do me a favour. (B)

Sounds wrong. Non-parallel tenses or aspects or something.

I asked them if they did me a favour. (C)

Sounds fine, but yes a different meaning. I asked them (in the past) whether (in the past) a favor had been done by them.

I asked them to do me a favour. (D)

Sounds fine. Indirect quote of "Do me a favor." or Please can you do me a favor?"

