Party Hatter is adorable! Especially when he was trying to put his nose through the wheel and then moping in the corner! And I've seen Alice lead him by the coat before! Haha!
@QwaiNine, I'm guessing that it's some sort of prop (like plastic cheese or something). It was broken so hatter thought it was okay to throw away. Alice wasn't so sure about that though so she kept the other broken piece just in case.
It's a nickname. The nicknames are for fans to identify face characters since there are multiple actors for the Disney characters. Like for Peter Pan, each of the actors have nicknames so fans can identify them. ie ; Filly Pan, Huggy Pan, and the legendary Spieling Pan. If you like this Alice, you search "Gingersnap Alice" because if you just search just "Alice" there will be many different ones. The actors aren't allowed to give out their real names so this is how we identify them. I'm basically repeating myself because I'm not explaining it well haha