Been a couple of years since I opened and used a Worldpay account, but yes they did require proof of trading and I had to get something stamped by the bank.
Didn't matter for me, I was registered as a sole trader.
No, they are a payment bureau, so process the transaction using their own merchant account.
I believe they can pay into any current account as long as it's held by the Worldpay account holder.
When I used Worldpay they were pretty good, but in recent years I've heard some stories of people been asked for unrealistic deposits before been issued an account.