Learn English – Why do we say a check “bounced”

We "check" with someone before coming and going in various circumstances. You mention a hospital; other examples include a hotel, a grocery store, a secure apartment complex. The expression comes from your identity, credentials, or authorization being examined ("checked") on your way in or out.

The expression has been variously extended to encompass other meanings nowadays: "I tried to get Joe to help, but he seems checked out these days" meaning that Joe is disengaged.

Your original question, before and after editing, exhibits yet another example of how subtle English can be; these two sentences mean very different things:

I checked out that hotel; it was awful!
I checked out of that hotel; it was awful!

The first implies you merely examined the hotel; the second implies you were staying there and decided to leave.

It is widely accepted that the others are all variations of F**ck All.

It is further thought that F**k All is in fact a misunderstanding of the phrase 'sweet F.A.', meaning 'nothing at all'.

The story:

The phrase 'Sweet F.A.' is of British military origin and refers to Fanny Adams, a girl who was murdered quite gruesomely in the mid-1800s.

British naval soldiers likened their unpleasant meat rations to the remains of Fanny Adams. 'Sweet Fanny Adams' or 'Sweet F.A.' was then applied as a slang term for mutton and eventually for anything worthless.

Later, F.A. was assumed, by those not knowing the origin, to mean 'F**k All'.

Finally, with the 'F word' being as harsh as it is, it is often replaced by euphemism or lesser expletives, which has given rise to the variations you mentioned.

Hope this helps.