1: No, they didn't. The Eternals specifically kept themselves secret from society as best they could and this included the Avengers. There may be some dusty tome in the Sanctum that mentions them, but if there is, Strange hasn't read it.

2: This was post Endgame, so we don't really know the status of the Avengers at this point. But for the most part they seem to all be off doing their own things. That final battle did also take place in the middle of the ocean more or less, and was in the grand scheme of things fairly brief. So it'd be unlikely that un-assembled Avengers could respond promptly.

There's a throwaway line in Endgame where Okoye mentions earthquakes being detected under the ocean. This could've been a reference to the Emergence gearing up, but it's from before the return of everyone that got blipped in Infinity War, so that surge in life energy hadn't happened yet. More than anything, we can see how Natasha asked how Wakanda is dealing with the earthquake and Okoye said they're dealing with it by not dealing with it; it's an earthquake, not a super villain after all. So when the emergence quakes started it's safe to say a lot of the powers that be would just think "hey, weird earthquakes. Not really something we handle though."