Month to month, 60 day notice rent

I'm not sure that's right. The whole point of a month to month lease is that it's only good month to month and either side can give notice within one month (30 days in NYC). Your landlord may include all sorts of provisions in the lease but that doesn't mean they will hold up.
If your deposit is same as your rent, there won't be any legal action (for what?) but why give your landlord that money? FYI - the landlord is required to give you back your deposit within a certain period of time and provide an itemized list of repairs if they use the deposit for something (forget what time period exactly) so keep that in mind too. And if they re-let your place as of November 1, they certainly can't keep your deposit and ask you to pay rent so something else to check. Finally, if they haven't been keeping it in escrow that's another basis to get your money back. NY has a lot of tenant protections. I think you'll be okay if you just do some research.

Doesn't matter when you moved out.
When did you give notice?
It's not 30 days, it's monthly. You need to give notice before the start of the month, as you pay a full month at a time, not for partial months ie days.
If you gave notice any time in September, then you are done at the end of October. But if you waited until October, then yes, you pay October, and you pay November.

fieldschester is correct, however depending on the wording it could actually be longer than that. If it is worded that you have to give 30 days notice of your intent not to renew for another month then a thirty day notice not to renew on September 10th would actually take effect October 10th which means you would not be renewing the November 1st holdover.

fieldschester is always correct and won't send you to live near the post office in west chelsea when you desire to live in the west village.

Except he is geographically challenged calling 7th Avenue "West Chelsea".

I don't think anyone cares.