I'm thinking of my neighbor taking my cat and expecting me to pay for food & vet bills when it doesn't even come over anymore
Why did your neighbor take your cat?
The cat didn't like to be with me sadly.
You should not let your neighbor take your pets. Keep your pets inside and take care of them or let the neighbor keep the pet.
Look at you, protecting all animals.
If you no longer want your pet you can forfeit them and let someone else take care of them. Sadly sometimes pets don't work out like peoples relationships. You should make sure they find a good home where they are compatible.