This is really a situation that requires one to look back on the lease or the contract that was initially signed between the renter and the landlord to verify justification.
Probably, depending on the terms of the lease. Many contracts require a full 30 days notice. If you leave before then, you still owe the full month's rent.
Yes, your contract obligates you for payments through a specific time. Even if you were granted an extension, it would likely be on a month to month basis only.