There is so many Omegaverse so it depends.
No exactly, this manga mixes historical stuff with a modern setting, so all is possible.
I think it all depends on the author. There are manga where they're in elementary or middle school before they took the test. Then there are manga that know before the child birth. I forgot the name of the manga, but it was set of twin of Alpha and Omega. The doctor warn the mothers before the twin were born that it be difficult to raise an Omega in a full house of alpha.

I have one question and correct me if I'm wrong but wasn't there like this thing that there was a certain age where you could find if you're an Omega Beta Alpha and stuff and you couldn't know just when you're born, like you had to be at least +10 years old??