Depend on what kind of game. Almost all game can be play offline. Unless you want to play online multiplayer game. I own PS Vita and PS4 too 

As long as you have download the PSN+ and your PS+ subscription still valid, you dont need to always online to play the game. It has expired date in the game itself following your subscription expired date.

Only two model

Model 1000 (First and the only LED screen - hard to find now, maybe not anymore)

Model 2000 (LCD screen and some better upgrade - im not into it by the way, still love my model 1000)