Teddy's jacket says the ship's hull number is DE-1025, which had been the hull number of the destroyer escort USS Bauer commissioned in 1957, decommissioned in 1973 and scrapped in 1974. It had no provision for a helicopter, however, as depicted here. The ships of this size and capability would later be called frigates in the U.S. Navy. At the reunion, the ship's name is revealed on the vets' jackets to be the U.S.S. Gertrude Stein.
Gene asks Teddy if he fell in love with a mermaid and she became human, but missed the ocean and eventually returned and he never got over it. This is the plot to Den Lille Havfrue, a Danish fairy tale written by Hans Christian Andersen in 1837.
Exterminator: Catch Me If You Ant
Store next door: I Wanna Spec You Up: Opticians
This refers to the song "I Wanna Sex You Up" by Color Me Badd from 1991.
The trumpeters are playing Barracuda by Heart.