How does one determine when to ask questions?

It is too vague to answer but usually people like it when you ask them good questions about what they are talking about or interested in, this is favourable to talking about yourself and own interests to them (unless they ask). Try to avoid asking obvious questions though

Your last sentence is an example of the bane of my social interactions. "Obvious," is a subjective term, what is obvious to one person is not obvious to another. There is also conflicting data; you say that people usually like good questions, and others have said that people do not like questions. They actually said "too many questions," but again, this is subjective, and thus useless in advice.

It is difficult not to be somewhat vague in my questions because I have trouble remembering exact details of conversations, and those conversations are in the past and (hopefully) won't be repeated, at least not word-for-word. 

I did not actually expect to get rules of conversation that I could use from this forum, but I recognize that I can be wrong, and so I asked the questions anyway. It appears to me that there are no actual rules, and that neuro-typical people have a sort of mind-minding. I know that it is not actual "mind-reading," but that phrase is useful to express my frustration. 

True. I have no idea if I'm being obvious or obscure, and stop to ask if people are following. You can bisect/triangulate where the dividing line is, I suppose, again by asking. 'Do you mind my asking...?'

What may be good is a few open-ended questions that allows the person to take charge of the conversation, rather than a series of clarifying questions that might come across like a police interrogation.

People actually say I'm a very good listener, although I have to restrain my pedantry, sarcasm and hair-splitting. Friends might be happy or upset about something and I often used to question factual basis of their opinions, which is definitely not what they wanted at that point.

Have you come across the idea of 'active listening'? It's basically paraphrasing what you've heard back to the person.; you may be right or wrong, but it can encourage someone to elaborate until some of the ambiguities are resolved. Can you walk away with unresolved ambiguities that may not be consequential?

It's more guidelines than rules, and the fact that most NTs aren't conscious of them, doesn't mean they aren't learned. They're just learned unexamined and practised so much most people don't need to consciously think what they are.  Autistic girls in particular often learn to blend in by observing what other people do and imitating. Some kind of symmetry where each person talks for about the same proportion of available time is a useful guide in any social situation, less so where there's an 'authority' figure.

True. I have no idea if I'm being obvious or obscure, and stop to ask if people are following. You can bisect/triangulate where the dividing line is, I suppose, again by asking. 'Do you mind my asking...?'

What may be good is a few open-ended questions that allows the person to take charge of the conversation, rather than a series of clarifying questions that might come across like a police interrogation.

People actually say I'm a very good listener, although I have to restrain my pedantry, sarcasm and hair-splitting. Friends might be happy or upset about something and I often used to question factual basis of their opinions, which is definitely not what they wanted at that point.

Have you come across the idea of 'active listening'? It's basically paraphrasing what you've heard back to the person.; you may be right or wrong, but it can encourage someone to elaborate until some of the ambiguities are resolved. Can you walk away with unresolved ambiguities that may not be consequential?

It's more guidelines than rules, and the fact that most NTs aren't conscious of them, doesn't mean they aren't learned. They're just learned unexamined and practised so much most people don't need to consciously think what they are.  Autistic girls in particular often learn to blend in by observing what other people do and imitating. Some kind of symmetry where each person talks for about the same proportion of available time is a useful guide in any social situation, less so where there's an 'authority' figure.