Month to month rental agreements require 20 day of notice before the end of the rental period. If I give notice on the 5th of October do I only have to pay rent for those 5 days, and can I move out anytime after proper notice has been given?

This depends on your lease. Generally, landlords like it if the rentals run from the 1st to the end of the month. So some will look at that and say that you held over for October, so you owe October's rent. Others will prorate and just charge for those 5 days. Quite obviously, the time to find out which your landlord does is before you give your notice. But nothing stops you from moving your stuff out before that date. I always overlapped my rentals, so that I could move my belongings to the new place and clean the old one when it was empty.

No, if you are required to give 20 days notice, then you need to pay up to the end of the rental period. Unless the landlord releases you.