You should be able to either contact your card provider or set your payment method to full balance each month so you don't have to worry about manually clearing it.
My statement date on the Amex is towards the end of the month but it gets taken out a week or two later (they control this automatically) but it always clears the full balance with DD.
If you want to get cash out of your CC without fees the only way I know of is to send friend/family paypal gift with the card as the source, it should be £0 fee, and then they can withdraw the cash and send back to you. Bit convoluted but I've done it a few times to essentially borrow money from myself