Are you allowed to carry the fish with you as carry on?
(in a styrofoam container)

or does it actually have to be stored underneath...?

(I've got this image of 'What about Bob')

But seriously, can you bring them on as carry on?

I have brought back fish and corals many times as a carry-on.

The rules say you have to have it in a leak-proof plastic container.

But I have had no problems bringing the fish in their plastic bags as carry-on.