at 14, you can work, but you won't be able to earn much....contact an area domestic violence program for help. you could potentially earn enough to get a nicer apartment that allows dogs.
if you can babysit for $5/hour cash 20 hours per week, that will be 400/month. some areas you may get more than that. you may be able to pick up dog walking business at night or house cleaning. you are supposed to report cash income to the IRS and pay taxes on it, but in reality at 14-17, no one will ever find out if you don't...just don't put a lot in the bank.
NOTE: if the house is owned, it takes YEARS to process a foreclosure. the process starts in about 3 months, but there are court hearings and delays...at worst it should be 6 months. if the house has room, a bedroom can be rented out...even if it means you and mom share a room.