Learn English – “check out”, “check in”, or just “check”

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).

You could say "My parents promised to buy me a new car" or you could say "My parents promised me a new car". Both are acceptable in the US.