just about anything as long as it has a lock of some kind counts as a "locked container" including gun rugs, soft sided containers, etc.
As far as entering a Federal forbidden zone, it doesn't matter whether it is locked up or not (state law). However, realistically if you have it locked iaw the state law and you are incidently in the area of a school, there isn't any reason to worry unless you do something stupid and it gets discovered. In other words aint nobody going to know it's there unless you get in a fire fight.
As far as National Parks the prohibition I believe is against "loaded" weapons, every time I've been in Nat'l Park I let the ranger know on entering that I have an unloaded pistol and I'm leaving out the other side and they always say just keep it unloaded.
The only time I actually lock the case down is when I'm going to be away from the vehicle.