How do I secure a vinyl window to the wall?
Yes put a nail into every hole.  I would recommend galvanized nails that are 1 to 2 inches in length.  If there is nothing behind the flange but plywood, anything over an inch and the nail will just be sticking through the plywood.  

I assume you will hit some framing behind the playwood, so a little longer and you will hit that.  I looked over my new vinyl windows in my sunroom and there are no attachment points on the inside of the window.  

I don't recall doing anything on the inside of the window when installing.  The most important point I remember was with the flashing on the outside.  Start at the bottom, then the sides, then the top so that water can be properly channelled away.  There is a video below in the comments showing an installation.  I remember it was pretty straight forward, just made sure the windows are level.