Online-only banks are real banks. They just don’t have storefront branches, which saves them a fortune in expenses, and therefore they can pay higher interest rates.

I have checking at a bank with branches local to me, plus I have a couple savings accounts at online banks. I’ve linked them, so I can transfer between checking and savings in a couple of days.

You can’t set up bill pay directly from savings, and they’re limited to six withdrawals per month, so yes, you would transfer money to checking and then pay bills from there.