Some sentences use "Aquí está" and some have "Aquí tiene" - meaning "Here (it) is". In what situation would one be used over the other?
They're mostly interchangeable, but in a sistuation like this "aquí tiene" is a little more specific to handing something over, instead of just presenting it. Not a hard rule though.
Cuenta is account,bill and sometimes invoice. It is NOT a check. You can check something, to verify, you can put a check on a suspension there are a dozen meanings. One is a cheque to PAY for something AFTER you receive the bill.
I'm not sure where your confusion is.
In American English, the thing you're handed at the end of a meal is often called a check. It is a synonym with bill. That piece of paper is called a cuenta. In every Mexican, Colombian, Ecuadorian, and Cuban restaurant I've been in, they call it "la cuenta" and it's what we call "the check".
This is not referring to a check that I write out (which draws money from my checking account). Some people in the US spell it cheque and it seems cheque is the only spelling used in the UK.
Because a receipt ("recibo") is the paper you get after paying, and not the check/bill itself.