Word Choice – ‘Anyone’ vs ‘Someone’: Which One to Use?

I think your distinctions between some and any sound correct, though I can't guarantee for certain that there are no exceptions.

Someone and anyone do not have a set rule, however. There isn't a hard-and-fast rule; they're used differently in different situations. In some situations there is a slight difference between them, in others I would consider them nearly interchangeable, and in others you must choose one or the other to be properly understood as they can have entirely disparate meanings.

John Lawler's answer here on ELU attempts to explain why there is no set rule for when to use someone and anyone:

You are correct. There is no clear way to do this. Robin Lakoff's paper entitled "Some Reasons Why There Can't Be some ~ any Rule" is precisely about this situation.

Short summary of a few of the reasons:

1. Any is a Negative Polarity Item, but some isn't.
2. Many environments (like questions) allow NPIs like any, but don't disallow some.
3. There are several kinds of any, including NPI any, and "Free Choice" any, as in
   • Any idiot can solve this problem.

And to answer the question in your title, it's idiomatic to use anyone there; "Is there anyone among you who knows the answer?" (that is, more than one of them could know it). So it should read:

Does anyone know the answer?

All three sentences are acceptable in both written and spoken English, but each has slightly different meanings.

Using "their" indicates you don't know either the gender or number of people who left the bag on the table.

Using "his or her" indicates you don't know what gender the person who left the bag was, but you know (or assume) it was only one person.

Using "his" indicates you assume the (singular) person who left the bag is male.

If I were to guess the most common spoken usage for this I would personally use the sentence "Someone left their bag on the table."