Am I required to give a 30 day notice if the landlord has no intention of trying to rent our apartment during that 30 days? They have already informed us that they intend to come in and do work on the place and that our deposit is 'nonrefundable', even though that was never part of the verbal agreement that we had with them. Only the pet deposit was to be nonrefundable. We have a dog but the people who lived here before had cats and you can still smell cat urine on the carpet in the basement, but they claim that is our fault as we did not make them put it in writing that was an issue. They are also supposed to be responsible for all yard work/maint. But they stopped mowing and snow removal 5 years ago, leaving it to us to take care of or not be done at all. Are we legally obligated to pay next months rent if we don't give the notice?

Yes, your obligation to give a 30 day notice typically isn't dependent upon the landlord's intent.

I think you are in the right about the deposit. If you can document that you paid them a deposit, the burden would be on them to establish that it was to be nonrefundable. A "nonrefundable deposit" is an oxymoron. The use of the word "deposit" itself implies that it is to be refundable if certain conditions are satisfied.

First ANY question on law needs to include your general locale as laws vary by area. A lot depends on what your lease says, but you are required to give notice of your intent to vacate. The amount of the notice will depend on your lease terms and the local laws. If you do not give notice, the landlord can hold you responsible for rental for the legal notice period.

You are required to pay rental up to the date you vacate. In some places you will be required to pay for the full rental period that ends after you vacate, i.e. your rental period is monthly but you vacate in mid month, you may be required to pay the full month.

If you paid a security deposit, then the landlord, generally, must return it or an itemized accounting of how it was used within a period dictated by local law. If you believe some of the charges are invalid, then you sue.