They both mean "thing" or refer to something, but there is a slight difference between them. In your examples, only the first one is correct: Omae wa shiawase (na) mono da.
While both mean "thing", mono (物) refers to physical things, while koto (事) refers to psychical/theoretical things (not material ones).
*Sono mono wa nan desu ka ? - Shorui desu. -> What's that thing ? - It's a document.(mono refers to the document, hence a physical thing)
*Nan no koto hanashiteru-n desu ka ? -> What are you talking about (here, koto refers to the subject of the discussion)
Hope this helps :D