Learn Italian: Hanno stampato il medesimo documento.

When placed before the noun, medesimo and stesso are interchangeable and both mean "the same". Medesimo is simply a more formal and slightly more literary synonym for stesso. Because this sentence talks about a formal item like a documento, this elevated register fits perfectly.

What would it mean if we put the adjective after the noun, saying il documento medesimo?

Moving medesimo (or stesso) to follow the noun completely changes its function. Instead of indicating identity ("the same"), it provides reflexive emphasis, meaning "itself". If the sentence were Hanno stampato il documento medesimo, it would mean "They printed the document itself" (for instance, as opposed to printing the email it was attached to).

Can medesimo be used on its own without repeating the word documento?

Yes! If the context is already clear from the conversation, medesimo can act as a pronoun. You could just say Hanno stampato il medesimo to mean "They printed the same one."