Their duty was to strafe the invasion beaches as far as I know.
What does it do ingame? Just FX, or does it actually hurt the soldiers on the beach?
Just FX, their duty was one thing, what happend is another. AFAIK IRL they didn't attack because AA fire from allied naval fleet was too strong so they withdrawn as soon as they arrived.
I wonder how the push will work (I guess push mode is implemented) - will it work like that: first three flags to cap and after that they're locked and Americans can attack other 3 flags?
yes.
EDIT: ok, my mistake, last flag is a separate sector.