Do the deep one rising tokens have to be placed if a gate is opening on a location that already have an open gate?

No, because in such a case, a gate is not being prevented from opening. Seems ad hoc, but that's the ruling.

So, what are the cases? Elder sign, Kate and?

That's essentially it. I think they wanted universal wording in case of any future mechanics, but I don't think there are any else.

I'm a little confused by this conversation. Are you talking about like a monster surge situation? So if a monster surge occurs, no token is placed on the Ancient One's doom track? Or am I talking about something totally different?

The Deep One Rising track is introduced in the Innsmouth Horror expansion, so, if you're still working on the base game, nothing to worry about (if you want the details, just ask; I thought not adding not useful details could be messy since you're learning the game)

In any case monster surges never add doom (unless you're playing with the Black Goat of the Woods Herald)

Ahhh ok, won't worry about it right now