I have to admit that I don't understand the Playstation TV at all. It was designed for the PS4 but it uses a Dualshock 3?
Well, it really wasn't designed for the PS4 originally. It's just a non-portable version of the Vita and the Vita can already do PS4 remote play so they decided to emphasize that. Still, I don't get why they didn't bundle it with a DS4 either...but I think they still work with it.
It will work with Playstation Now, but what about digital versions of PS3 games?
Not a chance. Totally different hardware and it simply isn't powerful enough. It's a Vita, not a PS3.
It plays most Vita titles, but what about Vita-compatible PS1/PSP titles?
Those should all work. They don't have the touch screen design issues that actual Vita titles can.