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The paper item "a cheque" is called "a check" in the US -- which presumably happened because "a cheque" has something to do with the verb "to check" (as in, "to check the health of"; to verify the status of something by asking/seeing it").

When you check-in to a hotel you aren't "handing over a cheque", you are checking-in with the reception. Ie., you are speaking to reception and checking your reservation/room.

To "check in with someone" is to have a chat or talk to a person to "check" how they are.

Absolutely. "In the UK you can pay a bill with a cheque, and in the US you can settle a check with a bill"

As someone who speaks the queen's English, I would like to recommend you look up what a cheque is.