Questions that come from contests should give a source. MSE has relatively strict policy on questions from ongoing contests, and clearly citing your source helps to ensure that you won't run afoul of that policy. A source for a problem also adds valuable context–it gives answers some idea of what the motivation for the problem is, and can give a sense of what kinds of tools should be expected (for example, an AMC 8 problem will require very different tools than a Putnam problem).
With respect to your list of questions:
- It is hard to say if it is "standard practice", because (as the answer to the next bit will show), not all questions tagged contest-math are problems from competitions, and because many new users don't know what the community standards about contest problems state. That being said, if a question comes from a contest, you should say what contest it comes from. If it doesn't come from a contest, you might mention that too. The fact that others fail to adhere to the standard is not an excuse not to adhere to the standard yourself (not that I am accusing you of this).
The contest-math description reads "Problems from or inspired by mathematics competitions. Questions regarding mathematics competitions," and the tag-wiki says
Suitable questions that would have this tag are of the following sort:
- Actual problems from mathematics competitions;
- Inquires about alternative proofs for a particular problem that is from a math contest;
- Questions that have been explicitly inspired by a contest problem;
- Soft questions requesting advice on competing in contests or those inquiring into the general utility of such things in terms of encouraging the further pursuit of mathematics.
See here for a list of mathematics competitions from which you can ask questions.
This seems to pretty explicitly indicate that questions that aren't directly taken from contests are appropriate. That being said, I think that the standard should be a little more than "This feels like it could have come from a competition." For a problem to be tagged contest-math, I would think that it should either come directly from a competition, or it should be inspired by a particular problem from a competition. In either case, you should be able to cite the contest you have in mind.
Eric Wofsey also suggests that "practice problems" might be appropriate fodder for this tag. Practice problems are likely more tenuously connected to specific contests, but even in this case I don't think it hurts to say something to the effect of "I am preparing for #CONTEST. This problem [comes from | is inspired by | etc] #SOURCE."
Unless you know that a problem comes from a particular competition, I would generally avoid retagging a question to add the contest-math tag. This question goes into more detail on this point.
As to you last question, yes, this is the right place for this discussion. Though it is recommended that you search first (for example, your point (3) has been discussed before (I linked to that discussion above). That topic was the fifth topic down when I searched "contest".