I have a question regarding the Omen. Let me give an example.

When you play Syzygy, you put Eldritch tokens on the red space, and add them whenever the Omen advances to the red space. However, there's a number of Mystic Ruins encounters, a blessing, and other things that let you move the Omen whereever you want without advancing the Doom.

If I moved the Omen to the red space as a result of those "put Omen, don't advance Doom" cards, would I need to add a token there? When that's happened before, I always move the Omen to the left blue space and avoid it, but I'd get more time if I could put it on the red space.

Does anyone know the answer for certain? I know Doom would never advance, but what about non-Doom related things.

You will not place a token on the omen track in this case.

Notice that the Syzygy card says to place a token when omen advances to the red space - the other examples you mention move the omen, but unless they use the specific term "advance" (and most, if not all of them do not), they do not trigger effects that happen by omen advancing.