Welcome to the forum. According to BT's Standard Residential Terms you get your first bill shortly after the service starts and it would have to be paid straight away. I assume it would comprise advance rental for the billing period plus any initial installation charges. They may also require a deposit or upfront payment but don't specify when that applies; you should be able to get relevant advice on 0800 100 400 (0330 1234 150 on a mobile).
I would opt for monthly billing and avoid the Monthly Payment Plan (quarterly bills with a regular monthly payment) as it is often more trouble than it's worth. Not paying by direct debit attracts a processing fee of £2 a month and paper bills are charged at £1.50 each.
Sorry, misunderstood. It is possible to rent phones from BT by calling 0800 800 150; see here. Few people do it these days as handsets are so cheap to buy.