What 4x4 lumber is appropriate for a skybed lounger swing?
Virtually any 4x4 lumber will be more than strong enough. Pressure-treated SPF would be fine, as would untreated pine/fir or cedar. You won't break a good-quality 4x4, as each is only carrying roughly a quarter of the load. 

Screws are almost unnecessary here, but almost anything that fits the holes well is fine. They carry very little load and simply prevent the thing from falling apart when being transported. 1/4" lag screws or heavy construction screws come to mind.