What causes the sudden spool-up sound from an F-16 when enabling afterburner?
My first guess based on the sound repeating when the thrust is idled near the end of the video is that's the sound of the actuators of the variable geometry nozzle.

Googling that actually auto-completed it to Great F-16 Actuator Sound (video).

Bonus answers to the bonus questions:


It's a ground test of the engine, making sure it's working fine with no leaks.
The F-16 has a tail-hook like carrier-borne planes, if you look closely a cable is attached to that hook, which itself is attached to the ground. The F-16, F-15, and other land-based fighters have it in case of an emergency, they'd land on a runway with a braking wire across it.