What can I do if I have been given a 30 day notice to quit (vacate the premises)?

My rent is due June 1st. Do I still have to pay it if they want me to leave 30 days from today May 22nd?
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In a month to month rental, they can give you 30 days to leave for any reason. If they gave you this notice anytime after May 1st, then you have till the last day of June to leave. If you are still there on July 1st, then they have to evict you. You will have no grounds to fight the eviction as they are merely asking you to leave and you haven't left.

Also, check your state laws. They might have to serve you a 60 days notice because of the number of years you were there even though you are a month to month tenant.
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Is it a court ordered eviction notice? In most states, you do not have to vacate the premises until it is so order by the courts. Many landlords and rental companies send out these "notices," but in all reality, its nothing more than a note you get from your landlord. They can not act based on that notice, and if they do, in most states you are entitled to your security deposit back, because they broke the law by illegally evicting you. Now, if it is court ordered, I would go ahead and pay the rent, and any refund due to you, will be mailed to you. Just remember, if it is not court ordered, you do not have to comply. Good Luck!!
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Eviction is the only way a property owner can force you out of your premises even if you have not paid your rent. You should not move only because the eviction procedure has started. You will lose the house only when the landlord wins the eviction action in the court. Even then the judge may give you extra time to move or Stay Of Execution if you pay what you owe, or you might be able to bargain for more time with your landlord.
If your landlord locks you out of your apartment or tries to force you out without a court judgment, he is breaking the law. Contact the police at once.
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Did you give them first and last month's rent when you moved in? (Many places require that). If you did, then you've already paid for the month of June and you're ok. Otherwise, you'll have to pay them for it when June 1st comes. So one way or another yes you have to pay rent for the month of June.
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Yes you will still owe June rent. A court order doesn't usually happen unless you stay past the 30 days notice you were given. Letting it go to court could buy you some time, but will be costly because your the one that will be paying the court cost
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Depends what your state rules are. Is there a long term lease or is it month to month? They can kick you out with 30 days notice if it's month to month. What's the reason for this? Are you behind on rent?
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Yes, you still owe rent during the time you live there (30 days from May 22nd)