Unless the Flip Effect also requires the Effect Monster to remain face-up on the field, then its activation requirement has been met (being flipped face-up) and will resolve unless something else negates it in the Chain.

If there are multiple Flip Effect monsters on Player A's field, then they will all activate and, Player A decides in which order they will resolve.

I would like to point out that optional effects that Flip Effect monsters have, such as Penguin Soldier, are placed later on the chain.

Turn Player Mandatory
Non-Turn Player Mandatory
Turn Player Optional
Non-Turn Player Optional

It's a form of priority that's been in the game from almost the beginning.

Source: http://yugioh.wikia.com/wiki/SEGOC

That page also goes into more detail of SEGOC mechanics.

Ah yes, there's that too.

So assuming Player B has nothing in the Chain as well, then mandatory Flip Effects will go off first followed by the optional ones, with Player A deciding how each one resolves in that order; first deciding mandatory, then optional, and resolving with the optional effects first as they are placed later in the Chain Link.