It is easier to do them underwater...

Handstands that is. The buoyancy makes your body lighter and stuff. I expect turbulence in the pool (water movements by others' motions or even bounced back from your own movements) may make it a little unstable (water can push you more than air, and carries ripples longer than air does) makes stabilization still pretty difficult (compared to just being lighter and doing a handstand in the air) but it would be easier on the muscles and stuff.

At the same time, you would really have to hyperextend your neck doing it underwater to keep your nostrils from filling with water. Was she doing this or maybe using noseplugs to keep this from happening?

Normally air trapped in the nose stops it from filling with water while submurged but only when upright, it gets harder when you're prone or supine and especially when upside down, it actually pulls the air right out and pulls the water right down into your nose.