it can't, because laeti has the same case and number as viri (Nom., Plur.) so it corresponds to a subject, thus it's an adjective and not an adverb, as in your case (BTW shouldn't it be happily instead of happy in your sentence?) Or you meant, that they are sleeping while being happy? I wonder how to convey that in Latin, but it's definitely different from the meaning of the proposed sentence.
Yes, I have seen 2-1-2 adjectives used before in some textbooks. Not all books use the same terminology. It's a little more specific than adjectives of three endings (at least in English) since some third declension adjective have three endings for each gender in the nominative singular (tend to see these referred to as 3rd declension three termination adjectives in English books).
Let me put my two cents in as well. I guess it might be useful to know that laetus doesn't exactly mean happy, it's rather merry , although they seem cognate, they are not the same (to me happiness is something more profound, more deep, not necessarily expressed through a visible gaiety - happiness loves calmness/silence; while you can pretend merry, or even sincerely feel merry, not being happy). Do you agree?
SO how would you say happy in Latin? I'm glad you asked. It would be felix (it has somewhat tricky declension, maybe that's the reason why they omit it at earlier lessons). BTW the name Felix (as well as Felicia , Felicity and Felicidad - both a Spanish word and a name) derives from the Latin word , meaning happy