What do these strange parts mean?

Solution 1:

Freezecam Taunt Appearances

Tracks the number of times you appear in enemy freezecams taunting while wearing the weapon. Similar to taunting achievements, I believe you have to be the focus of the killcam (the killer), not just in the background.

Full Moon Kills

It means a literal (real life) full moon. Full moons come once a month (give or take), so check your Lunar calendar for the date when a full moon will occur. On a related note, full moons coincide with the ability to equip Halloween items outside of Halloween (for those days under the full moon only).

Airborne Enemies Killed

'Airborne' literally means 'transported via the air', which leads me to believe it means any character that isn't touching the ground. I don't see why they would exclude a jumping character over one that was rocket-blasted or airblasted for example.

Point-Blank Kills

I daresay they're using the more common definition of point-blank, (Not the one concerned with bullet drop off) i.e. within 1m (3ft). From personal experience you basically have to be bumping into the other player to get points towards this.

Allied Healing Done

Adding this Strange Part to a Strange-quality weapon will enable it to track the total number of allied health points you directly restore with that weapon.

In other words, Sandvich, Health Packs etc do not count for this part. (Unless you can get a Strange Sandvich, in which case I want one)

You can see a list of all the Strange Parts on the TF2 Wiki. If you hover over the names, a tooltip will appear with a short description of that item.