PS4 is kind of like that. When you install the game initially, after maybe a 4GB download, you can play for a while. But then you have to wait for the rest of the game to install (sometimes 10-40GB more). For example, Lego Batman 3 let you play one level before it said you can't go any further. Star Wars Battlefront just let you run around as Darth Vader killing people for a while.
When it comes to updating, you can play without updates but they're usually installed while the system is in stand by. If you haven't played the game for a while, it may not do it automatically but will start next time you play.
And, ouch, bought a game on a deleted profile. They like to screw people with that shit. I have games linked to 3 different accounts on my PS3 and PS4, and I can't unify them because they've all got content I want to keep.
Games with Gold is getting semi-decent games at least. The free games with PlayStation Plus have been just awful since the PS4 was released. I don't even bother trying them out anymore.