Gidget is in love with Max, who is oblivious and uninterested... because he's probably been neutered. He taught one of the other dogs how to "sit the comfortable way" after getting fixed, which means he probably had the same experience. Plus, it's Gidget who brings it up.
From that, one could say Gidget is unaltered.
Or that, whether she is or not, she's simply been watching that telenovela way, way too much.
Duke's commandeering of Max's dog bed seems really, really mean, especially since it's way too small for Duke to fit into and he has to drape himself over the sides. But the flashback to Duke's life with Fred shows that he'd done the exact same thing when he and his owner took naps together, draping himself over the armchair's arms and Fred's lap. So lying on the too-small bed was the closest Duke could get to being with Fred again.
It also makes sense that Duke rescued Max from drowning so easily, as he's a Newfoundland, a breed which has been known to save drowners in Real Life
Duke's owner, Fred, is explained to have died. Duke did not know this because his owner did not come to pick him up - and he was left at the pound. Fred was already pretty elderly when he was shown in the flashbacks. There's a pretty good chance that Duke could have been the only thing keeping Fred going - and Fred's version of the tale is that he ran away chasing a butterfly and got lost, so for all we know, the last thing Duke heard from Fred may have been a "Come back boy" or "Be good".
In the same vein, there's a good chance, given that the setting is New York City, that some of the Flushed Pets weren't abandoned, but lost and separated from owners who never saw them, again.