What flags are displayed here exactly? And to the right, what version of (what I assume is a Gadsden flag) is that?
The other flag is the flag of the Italian Navy, where you can see the banners of the 4 great Maritime Republics of the Middle Ages: Venice, Genua, Pisa and Amalfi.
The Gadsden flag was an early American revolutionary symbol, from a political cartoon back when all the colonies needed convincing to unite.
Because of the post-Bush Tea Party movement trying to claim the mantle of the revolution, it's more recently fairly ubiquitous in America on the bumper stickers and vanity license plates of people advertising how right-wing their views are, and is divorced from its original meaning.
Almost right. That of the link is the flag of the civilian ships. On the flag of the picture there is instead the symbol of the Italian Navy (it has the Naval Crown over the symbols of the Maritime Republics). https://it.wikipedia.org/wiki/Stemma_della_Marina_Militare