There are any number of reasons the landing gear could have come down. The primary landing gear handle is not directly mechanically connected, however the alternate gear extension lever is a mechanical connection so if that connection were pulled as the cockpit separated, it could have activated the emergency landing gear extension mechanism.
edit - losing utility hydraulics should not automatically lower the landing gear. There are mechanical uplocks that would not automatically release with simple loss of utility hydraulic pressure. That said, the hydraulic system is very complex and tearing off the nose, which has hydraulic components, could have weird results due to either hydraulic system effects or other mechanical issues with the various actuation systems that run from the cockpit to the rest of the plane.
But really it could be almost anything and without reading the actual engineering analysis, it's just speculation.
As for the maneuvers, it appeared to be a standard BFM setup. The beeping you heard was either an angle of attack or G limit warning (not sure, I never heard what the F-15C tones sound like) so there was some maneuvering going on at the time.
I was a bit suprised to see the nose ripping off upwards rather than downwards, but a failure of one component while pulling G's might not result in the nose ripping off until the G's were relaxed if the failed part was not required to hold the nose on while pulling G's, but was actually required to hold the nose on during lower G flight. A part that is under tension while pulling G's and under compression when not pulling G's (or vice-versa), could certainly fail under G loading but not result in a catastrophic breakup until the G's were reduced.
