I used the Adobe Illustrator file with template blank layers.
1. when my puppet blinks in CA its head gets squished and it also lifts off the ground. What can I look into to fix this?
2. What can I do to get movement in the hair of the puppet when I sway back and forth?
Usually it’s a sign something is not marked as independent. I wrote up most problems I have seen with eyes at https://extra-ordinary.tv/2018/04/21/debugging-character-animator-eyess/
For moving hair you create an independent layer, attach near top of hair to Head, then use the dangle tool to add a dangle point near bottom of hair. (Search for “dangles” to find more details.)
would you use the dangle tool to make a mustache or beard also move?
Sure. Clothes too. it adds a bit of dynamic feel to the character.