Agree with pmusser - they're both correct.
However any is a Negative Polarity Item, whereas some is not. Also, when you say "Has someone seen my bag" you actually mean at least one from the group has seen your bag. On the other hand, "Has anyone seen my bag" is a general addressing.
To make 'someone/anyone' clearer, think about these sentences -
Someone can do it - at least one person can do it.
Anyone can do it - any person (or better everyone) can do it.
More examples as you asked are here. See Travis' answer in the same, which says:
In my opinion, the big difference between "someone" and "anyone" is that "someone" refers to some person, and that person is specific, even though it may not be known, while "anyone" refers to some person, and all people are equally interchangeable as said individual.
To begin with, the second option is not possible here. The formula "his or her" is reserved for generic subjects, when you don't know or intend to specify the sex.
That person who just left forgot his or her coat.
A teacher should respect his or her students.
So is the first one okay? More or less. In most settings, you would use "their" in a case like this.
In higher registers (maybe as high as an academic paper), you would avoid "their" in this sentence. The solution would depend on the context. For example, if you're talking about the same class and the same assignment, you would probably say "the":
Neither John nor Mary has done the homework.
But if you couldn't do that, you might have to reword entirely:
John and Mary have not done their homework.
That said, be aware that the correct use of "Neither ... nor" itself suggests a higher register. Hence, it doesn't jive very well with this use of "their". So you might reword it in a casual setting too.
One way of capturing the emphasis that "Neither ... nor" gives in a casual setting might be:
John and Mary both haven't done their homework!
Best 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."