(1) Would you mind if I smoked here?
Is (1) a conditional (i.e., a sentence containing a conditional clause)?
Or is the if-clause complement of the verb mind?
EDIT
I feel the need to make it clear that the intended context for the OP is what you normally would expect, i.e., a question coming from a smoker who'd like to smoke at the time of speaking. Somehow I feel that this type of question is not to be lumped together with a hypothetical question where the speaker does not necessarily intend to smoke at the time of speaking.
Best Answer
Yes, it's interrogative and conditional, with the condition given in the "if…" clause (protasis) and potential consequence in the first clause (apodosis) and inversion ("would you" instead of "you would") to make it a question asking if this conditional is indeed the case.
To answer questions from the comments:
"Would you mind…" makes it more immediately clear that you are starting a question, "If I smoked here, would you mind?" remains valid too though, and also interrogative and conditional, but making it clear from the beginning that one is asking a question is often an advantage.
It's normal with this sort of conditional to use the past tense. Consider if we make the statement, rather than the question; "You would mind, if I smoked". This sets up a hypothetical situation ("I smoked") and then makes a statement about that depends upon it being true ("You would mind") even though it is not true (this is the counter-factual conditional form; about things that are not true, and what would be consequent upon it if it was true).
Historically, the tense used here is in fact the past tense (simple past, as here, or past progressive) in the subjunctive mood. The subjunctive mood isn't really distinguishable from other moods in modern English for the most part, but it can be seen in the variant ("You would mind if I were to smoke") as the subjunctive uses "I were" rather than "I was". That said, many people ignore this now and so you might also find "I was" used here by some native speakers.
It makes a reasonable amount of sense, when one considers that it hypothesises one event coming before another; first "I smoked" and now "you mind".
And hence just as when it is not a question it is "you would mind if I smoked" (or "if I smoked, you would mind"), so as a question it is "would you mind if I smoked?" (or "if I smoked, you would mind?" or "would you mind if I were to smoke?"). The question hypothesises one event coming first ("I smoked") and then asked whether it would then be true that now "you mind".
All that said, you certainly will come across people using the full present-tense "would you mind if I smoke". Whether that is "correct" or not is one of those things people will disagree on.