Learn English – Using “I am writing to…” with “request”

Do the needful is Indian English, which has been covered on ELU.

If you're only interacting with other speakers of Indian English then feel free to use it, but avoid it in any other contexts (most Americans and Brits will think it's quaint/uneducated).

In general, the "standard" form is do what[ever] is necessary, but in OP's specific context most likely nothing like that would be used anyway. If you've just asked for an email address, it goes without saying that you want the other person to do whatever is necessary to give you that information.

I may be wrong, but I have the impression that for many Indian English speakers, "Please do the needful" carries a subtext of "This problem is too complex for me to understand or resolve myself, but I have complete faith that you will be able to deal with it, because you are very skilled in such matters"

As I said, Brits and Americans wouldn't normally use any equivalent for such a trivial problem as finding someone's email address. But if the request was for something more challenging (and crucially, if it was from a manager to a more junior worker), "Do what[ever] [you think] is necessary [to solve this problem]" might be perfectly normal. The implication there is that the manager is authorising the junior to do things he might otherwise not be "permitted" to do (in effect, the junior is being temporarily "promoted" for the duration of the problem-solving).

In that context, it should be clear that (to Americans or Brits, at least) any such phrase would probably be considered offensive/cheeky if addressed to an equal in the workplace (if the person asking isn't senior enough to confer temporary authority on you, they shouldn't be speaking to you that way).

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).