So, I was downloading Infamous: First Light on the PS4 and the total size it says "9.6gb" to my surprise at 6 and something I could start the game, but some "data app" has started downloading. My question is, is there a problem if I start the game like that? will I have performances troubles and stuff? I´ve been reading and this looks normal in some games, like The Last of us but I´m not sure if is more like "to test" the game or to play as normal.
This behaviour is completely possible. According to the Features page of playstation.com:
Play as You Download
Increase your play time with the ability to play games while they’re downloading titles. When a player purchases a game that is enabled with this feature, the PS4™ system downloads a portion of the game, so play can start, while the rest is downloaded in the background during actual gameplay.
When a certain part of the game is downloaded, you can play. You can only play a certain part of the game, as pointed out by Nitro.de. In case of DA:Inquisition you can only play until the Tutorial Boss. You are more likely to hit the limit when your download is slow.