Question: Do the main characters have to have a last name and should those characters backgrounds (careers, hobbies, etc) be described at some point in the story, and if so where or when? (Early on in the story or at any point in the story.)Answer:
There are no rules about these things. Giving your character a last name usually is a way to make them seem more tangible and real, just like with any other trait you choose to give them.
Sometimes you will see books that are very introspective and character-driven in which the main character's name may be absent or reduced to a single name. The effect is to disassociate them a little from the kind of external world environment one would normally expect.
Often, a character is referred to primarily by their last name, especially in an action-driven, adult story. But you may see first names used primarily in children's or YA books.
Character backgrounds, like other traits, should appear when they are relevant to the story or when you want to make the main character seem more real to the reader or to build a connection or rapport between character and reader.
Again, there are no rules about when character background should appear. It all depends on your story.
Often a little information in the first chapter helps to build the connection between reader and character, but you don't want to create an infodump either. You may be better off choosing one particular detail or incident to relate that is indicative of the character's personality or approach to life rather than rattling off a lot of details that are trivial.
Other information about the character may be presented throughout the book when it's relevant.