Learn English – What do you say to a guest when they are departing your country and you are accompanying them e.g. to the airport

Checking something out can mean "taking a look at something" when used colloquially. So with the first three sentences, checking out does not necessarily mean that you're going to eat there.

If you say "I am going to check that restaurant out" it can both mean "I am going to read something about them, get myself informed" or "I'm going there to try their food". However it usually means the latter, but that's not a rule.

Try out / try is fairly interchangeable in your example sentences, although it sounds better without the "out" to my ears.

Situation 1:

You would say "Have you checked your bag?", not "Have you checked out your bag?".

And yes, that's another way of saying "Have you looked in your bag?"

Situation 2:

"Could you check your inbox?" is colloquial but it would work. As would "Could you [please] check your email [as soon as possible]?". The word "mailbox" however can't really be used when talking about an email as that word is commonly used to refer to a physical mailbox (The one in front of your house).

While the above answers do make sense they come off as blaming the other person.
I would instead try something like

I think there has been some miscommunication.

This way no one is being blamed but the issue is still raised.