I got a termination notice in the middle of the month and my rent is due on the first. Do I have to pay a pro-rated rent from the first until the 13th of the next month?

If you or your stuff are still on the premises, the landlord will charge you by the day.

Seems like the landlord wants you out by the first of the month....soooo you still have to pay for the month you stayed there but if out by the first of next month you wont owe for that month. If you aren't out by the first you may be charged for that month too. Talk to your landlord and work it out.