Nearby would also work. Is there a bank nearby?
Yes, this is now accepted.
"Is there a bank close by?" should also be accepted.
Does 附近 mean "around" as well as "near"?
It can also act as a noun, meaning "vicinity."
Yes, I'd translate it to "nearby" here (this is accepted).
Is this correct english?
Yes, it is. It sounds like American English to me, I'd translate it as ”nearby".
"Is there a bank round here?" should be accepted; plenty of people shorten "around" to "round". Said use of "round" is perhaps slightly informal, but it's too common to justify not accepting it here.
Eh.. kind of. "Round" is just an abbreviation of sorts. When written, "round" meaning "around" is written as 'round