It depends on the context. If a group of people is brainstorming (that is, trying to come up with a bunch of creative possibilities to solve a problem), then I might ask someone "Do you have any ideas?" In this situation, I am expecting that they may have several ideas that are relevant to the problem.
However, if I am asking for an answer to a specific question, I might say something like, "I can't remember what the capital of Pakistan is, do you have any idea?" In this situation, I am expecting that they might have one specific piece of information to offer. This is probably a more colloquial or idiomatic expression than the other one.
And yes, the use of "any" does have a lot to do with questions and negatives. "Any" is a negative polarity item, which means it can only happen in certain contexts. For example, I would never say "I have any ideas" or "I have any books", but I would say "I don't have any ideas" and "I don't have any books". I don't really know how to give you a full explanation of where or why "any" can be used though. The example you give about borrowing "any book" is certainly correct, even though it isn't negative or a question.
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