Not quite. "At the latest" and "at the earliest" mean the same thing as "no later than" and "no earlier than". They suggest that there is a deadline, that something can or will not happen before or after X. For example: Please submit your application form by February 20 at the latest. I expect to finish the project on Monday afternoon at the earliest. The expressions "as late as" and "as early as" are used to emphasize that a particular time or period is surprisingly late or early. For example: Some girls get their period as early as age 9. Nowadays, adulthood can start as late as 30. A sentence like, "the package could arrive as early as Monday" suggests that it would be surprising to have it arrive so early but it might happen. While "the package will arrive on Monday at the earliest" means that it is certain that it will not arrive before Monday.

Yes, apart from where you mention the specific time. With the first version, you would say the time before, whereas with the second alternative, you would mention the time afterwards. ie. ...tomorrow at the latest, or ...as late as tomorrow.