They are both correct.

Both are OK! We don't say "post" in the U.S. but they say it in the U.K. and other places.

Both are grammatically correct. You could also use mailbox or postbox. or postage box. However, the word used most often is really dependent on local custom. For the part of the United States where I live, we would say "This came in today's mail." Some might say "This was in the mailbox today." But, the first example is the common answer. In the United States, I am not aware of people using "post" by itself. I am sure other English speaking countries have different answers!