Thanks for playing! I'm the developer of this game, and I agree that it needs an FAQ! There is all kinds of hidden stuff in Alien Squatter, and I wonder if anyone has ever found it.
Concerning the pets: You can only have one pet at a time, so if you already have a pet, you'll have to get rid of it or the opportunities to get a new pet won't work. If I recall correctly, you can get rid of a pet by selling it to the curry shop in the northeast corner of the map.
You can get a pig by repeatedly buying curry from the curry shop. Randomly, when you buy a bowl, there will be a pig in it.
You can get the cat by collecting roses. The more roses you have, the more likely it is that the cat will appear in your tent at night. Then you can adopt her.
You mentioned the chinchilla already.
There is a parrot too. I can't remember how you get him.
There is a robotic fish. I think you can buy it from the junk shop on the north side of the map.