Jess walked into a restaurant in Madras, India. Only one booth was occupied by
a group of teenagers who looked like they'd been there hours waiting for someone.
Waiters and cashiers weren't anywhere to be seen. Suddenly, [a/the] door behind the
counter opened and came out a waiter carrying a tray with a hot pot of curry.

Are they any mistakes? Also, which article is correct?

a door: when you mean one door of many existing behind the counter
the door: when there's just one
the cashier: I don't think there's more than one, usually
a waiter came out
out came a waiter: he popped up surprisingly
Thanks MH. I get it now.