Learn English – Difference or similarity between today, these days and nowadays

In this case, "try" and "try out" both mean "to test/examine".

You have to look at what comes after the "try":

1. try + noun = test: Shall we try the new restaurant?
2. try out + noun = test: I want to try out the new Mercedes.
3. try out + for = attempt: I'm going to try out for the football team.
4. try + on = test clothing/accessories: Try on this perfume and see if you like it.
5. try + infinitive = attempt: Let's try to get seats close to the stage.

Check your question to see how I edited your example questions.

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).