Hi Elizabeth Yes, I'd say 'hatched out' is OK. "we saw some baby birds that hatched out of the eggs" "we saw some baby birds that hatched from the eggs" "we saw some baby birds, hatched from the eggs" "we saw some newly-hatched baby birds"

Yes, "hatched out" is all good. Since "the eggs" is plural and definite, "the baby birds" also needs to be plural and definite to match. You could also use "some baby birds", as Neal suggests.