I can't speak for the Hornet, but I'm sure it has a similar pressurization schedule as the Hawk.
Pressurization starts at 5000 ft MSL (below that, we are at the same pressure as the air surrounding the airplane). Our max delta P is 4 PSI at 40 000 ft. After that, it maintains 4 PSI.
So, our px schedule is as follow :
Aircraft altitude Cabin Altitude
5 000 ft 5 000 ft
10 000 ft 7 000 ft
20 000 ft 12 700 ft
30 000 ft 16 800 ft
40 000 ft 20 100 ft
As a rough guide to see if we're pressurizing correctly, we use (indicated altitude divided by 2 plus 2000 ft).
The risk of not being pressurized is pretty big. We can suffer from decompression sickness (of the bends). It can be very serious and may cause death, in extreme, untreated cases.
The way the cockpit is pressurized is pretty simple. Air from the compressor is taken and blown into the cockpit. There is a pressure controller that regulates the rate at which the air is let out of the cockpit via the two valve (one in the nose, one in the back of the cockpit). There are other valvles preventing over or under pressurization of the cockpit.