Yes.

The Mayor is permitted to speak after the Magic Word has been revealed, and can add any input into who to vote for (or defend themselves if being accused).

Technically, yes, they need to give that player the Correct token. However, it is well within the spirit of the game for a Werewolf-Mayor to "not hear" a guess if they can get away with it (for example, multiple people are talking at once, and they opt to respond to the other person).

No.

The Mayor usually wants the Village team to win, and should answer questions in a manner that leads them to the correct answer. Unless the Mayor is a Werewolf, they should be answering the questions with the goal of giving players more information that will help them figure out the magic word. A good example of an answer that is technically a lie (but a valid response from the Mayor) is the question "Is it an animal?" when the answer is a human. Even though humans are animals and the truly correct answer is "Yes", the Mayor (an innocent Villager) knows that would be misleading to the Villager team and should answer "No". However, a Werewolf Mayor would be able to answer "Yes" to that (because they're allowed to mislead and lie to players), and then defend themselves on the basis of it being technically correct, even though it would lead the guesses down the wrong path.

Werewolf and Minion. The Doppelganger may or may not be depending on their Night Phase action.

Villager, Seer, Apprentice, Mason, Thing, Beholder, Fortune Teller. The Doppelganger may or may not be depending on their Night Phase action.

No.

The cards act functionally the same in the game and purely have different art.

Yes.

If you run out of tokens as the Mayor, you are unable to provide answers to more questions. The Werewolf team can use this to their advantage by asking a lot of questions that lead nowhere and running the Mayor out of answers.

If you have freshly installed the app, it will default to showing all the available roles for the Standard Edition of Werewords. Click the gear symbol on the app main page, and select the box art of the Deluxe Edition to see all available roles.