1) When using the Ballroom, may a bewitched player deliberately fail to say "Thanks, your Excellency" in order to deprive the Witch of the use of her ability? Or does the Witch still get to do her part of the player's turn? It seems a bit twisted to cheat the Witch out of her power.
Would say no, although I would also expect the Witch to say "Thanks, Your Excellency" when her turn first began (when number 1 was called) or else she wouldn't get to use the character's power now. Strictly speaking the bewitched player's turn ends as soon as they take an action, and the witch's turn "resumes" with the use of that character's power.
2) When the Witch bewitches the Emperor, does the player of the Emperor move the crown, or does the Witch? It seems to me that the Emperor does since this ability happens as the Emperor is revealed, yet it is a specific power of Emperor and Witch is supposed to get to use the bewitched player's powers.
I would say the Witch player moves the crown, however, the Emperor player and the player who currently has the crown remain the two invalid targets for new ownership. This could well mean that the Witch can give herself the crown, if she wants to. I make that ruling based on the portion of the Witch's entry that says the King still gets the crown if she bewitches the King.
3) Must you use your character's power? Specifically, can a player the Witch card in order to go first, and then choose not to bewitch somebody so that she can build on her own turn, thus becoming the first person to get 8 districts?
The section of the rulebook on character powers says "may use" so I would say it's not mandatory. The Witch could choose not to bewitch anyone and instead act first.
As an aside, the Witch strikes me as a mostly useless character and we rarely use her ourselves. The only time she's handy is if the character you want is already gone by the time your pick comes around, and you're just desperate enough to gamble that someone else has him and he's not simply out of play. Most of the time if the character I wanted is already gone, I can make do with someone else.
If the player the Witch chooses to "bewitch" chooses to take 2 Gold Pieces for his action, does he get the 2 gold, or does the Witch get it ?
The original player gets the gold (or cards.) The player's turn ends after taking an action (ie: choosing gold or cards) and the Witch's turn begins from that point forward.