Dot the i's and cross the t's before you stop paying rent. Have you notified your landlord about the heater by certified mail, or has a government agency issued a violation notice to him? If you can prove he knows about it, and it presents a health, life or safety hazard, you may be able to get a court order to pay rent into an escrow account rather than to the landlord. Or you may stop paying, and use "no hot water" as a defense when he tries to recover the rent.
A landlord can keep all or part of a security deposit for unpaid rent, damage due to breach of lease-that is, costs to re-rent the apartment if you move out before the lease is up-or damage to the unit beyond ordinary wear and tear, such as your cat scratching the door frames or you spilling wine on the carpet. But if you paid the rent, left when the lease ended with proper notice, and the place was clean and undamaged, a landlord who doesn't return your deposit may be liable for up to three times the amount withheld plus reasonable attorney's fees.
Possibly, but you'll have to spend some time in rent court. You can file for repossession of the apartment and for the overdue rent and court costs. She has the right to stay by paying all amounts due before being evicted, but loses that right if she has racked up three or more judgments for possession for unpaid rent in the preceding 12 months.