The TVTropes Trope Finder is where you can come to ask questions like "Do we have this one?" and "What's the trope about...?" Trying to rediscover a long lost show or other medium but need a little help? Head to Media Finder and try your luck there. Want to propose a new trope? You should be over at You Know, That Thing Where.
The first is discussed in Musical World Hypotheses, because this is something determined on a case-by-case basis.
As for the second, you are asking about Video Game Interface Elements. There is a trope for the opposite side — if the characters can see their HUD, then it's a Diegetic Interface. I'm not sure that there's a trope for non-diegetic interfaces as this strikes me as Too Universal. How interesting would it be to have a list of games that use interface elements?
@ KD I thought No Trope Is Too Common?
This is kind of hard to explain so I'll just give 2 examples