As far as the kids being allergic to dogs, that is what allergy medicine is for, DUH!!
I am deathly allergic to rats, my daughter keeps three for pets and I feed 8 of my 12 snakes rats. I buy generic 24-hour allergy medicine at Wal-Mart for around $5.00 a bottle, it lasts me 3-4 months and keeps me symptom free. Don't let them use that as an excuse!
Now, as to your so-called mom and her boyfriend. Ask him once to help cover the vet bills, if he tells you to sod-off, and I'm sure he will, tell them both you'll be filing charges against them both with Animal Welfare for neglect and cruelty, and you'll see them in court if they continue to refuse to help. Then DO IT.
I understand about making the best of a bad situation, believe me. But that does not mean letting them use you for a doormat, which they will if they think they can. I won't bore you with details, but let me just say, there came a time in my life when I got tired of being stepped on, and when I snapped, I REALLY snapped. But you know what ? My life is 1000 times better now, and the people who treated me that way get along better with me now than they ever did before.
I wish you the best.
Gale