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In this weather, the failure to provide heat is a major violation of the landlord's obligation. The other things you mentioned sound like problems that have existed for a while. I suggest you get the entire list together to the landlord and send it with a demand that the problems be remedied in a short period of time. Make clear that if they are not you will consider it a breach of the lease and will take steps to relocate immediately rather than at the end of the lease. I have assumed that you have not agreed to renew the lease.
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Yes you can but rather than just leave if they are not repaired, you can take the landlord to court and seek a declaratory judgment that you can terminate the lease without penalty.