Prison just sitting there

Part of your problem appears to be that your Canteen is not designated as a Canteen room type, so as far as the game is concerned you have a big room with tables benches and serving tables next to the kitchen, but no Canteen. The cooks aren't cooking anything because they don't have a Canteen to cook for. Likewise the inmates are not eating because they don't have a Canteen to get food from.

As for them not going to the Yard, Showers, Common Room, etc during free time make sure that there is a path from their cells to those areas that does not require them to go through Staff Only areas. If they have to go through Staff Only areas they need to wait for a guard to escort them, if the free-time period is too short that might not happen before the period is over.