What counts as plagiarism?
Probably the question cannot really be about plagiarism, literally, because most traditional senses of "plagiarism" refer to specific choices of language, rather than "ideas".

In the first three cases, Bob's behavior would be completely within the range of standard operating procedure. Public information is public. Yet, also, at the same time, many people would consider such behavior rude and mercenary. That's not the same as illegal or unethical, whether or not it is immoral.

(The fourth case is harder for me to simulate in my head, and I have no comment on it.)

Yes, one could imagine that mentioning a problem or idea publicly gives one some sort of right to it. However, in practice, this cannot possibly work, because it is extremely difficult to gauge what more is required to make an idea into something substantial. The only viable default is to imagine that whatever one says publicly is then truly available for all. One cannot control it, or guarantee "getting credit". Yes, the most virtuous people will give credit to the origins of their ideas, etc. ("Duh!")

If the real question is how to stake a claim on ideas in a fashion that prohibits others from doing anything with them ... as opposed to taking the idea, doing something, and giving credit for origins... I have no suggestion except keeping secrets.

EDIT: and, no, merely expressing an idea does not create "possession"... Not at all. But, among decent people, credit would be given. Can't rely on this, etc.