I'm unclear on when you can play Halley's comet:
-Presumably you can play it on the targeted player's turn, but does it have to be on the turn you want to effect (when you force them to skip or discard), or on their previous turn only?
Previous turn only. If you are playing on the target's turn, his next turn referred on the card is, well, his next turn. But this is not really a problem due to the next two answers.
-Can you play it on one player's turn to affect the next player? E.g. Hapsburgs play it against the Ottomans, in response to the Protestant play
Certainly. There's nothing saying that you have to target the player that is currently taking a turn.
-Can you play it on your turn in response to your play of another card? E.g. England plays a card to move to Paris, then plays Halley to skip the French turn
Yes. From 6.1 in rules: "Response cards are playable as an event in the Action Phase during any player’s impulse (including your own). Play of a Response card interrupts a player’s impulse, a battle, or an Event card play." So, you may interrupt your own card with your response card.