Have only read OP's posts, so I expect I'm just repeating what others have said.
I can honestly say - STOP visiting them. The add nothing to your life, quite the opposite they give you nothing but stress.
As for "I went because my eldest loves my older sister who lives there and loves seeing her" - he's three. He loves everybody. Stop going there and after a few weeks he will have other people / things to love.
You are the adult here, and it is your job to protect your children. It would be lovely if your mother/sister were worth a damn, but they're not. You need to stop going to their house to protect your children from the dog, and to protect your children from your mother/sister. Just as you noticed there are no pictures of your children on the walls, so will they (eventually). Or they'll have just felt the lack of love from them. Don't subject your children to that, surround them with loving people.
You don't mention your father at all, except to say that he lives with your mother/sister. And there's another sister? What's your relationship like with them?
But, main point - don't go to their house again. Invite them to yours by all means (safe in the knowledge that they won't).
And DO NOT pay the vet's bill. That came about because they have failed to train their dog, and that is on them.