A: You get as many as they can give you that they’re supposed to. If that’s none, then that’s the way it goes.
A: Yes, you must complete your compound Action (Draw Two and Use ‘Em, Draw 3 Play 2 of Them, or Fizzbin) before you opt to take any optional Free Actions allowed by the Rules.
And, just to preemptively respond about Keeper powers as well, Andy rules that those should also be held until after any compound Action is resolved.
Example: If you could win with the current Goal if you had a Crew Member you don’t have, and you Draw 2 and Use ‘Em, receiving the Captain and some new Goal, you can’t play the Captain and steal a Crew Member to win with the current Goal before you complete the compound Action by playing the new Goal that you can’t win with. Them’s the breaks sometimes.
A: The first answer is very easy: No. D3P2 does not technically go in the discard pile until you are completely done executing everything on the card.
You also seem to be asking whether the card you don’t play from D3P2 is in the discard pile, available to pull out and use with Let’s Keep Doing That.
Technically, you should execute the instructions on D3P2 in the order stated: Play 2 of them, and [then] discard the last card.
So you play D3P2. It’s not technically in the discard pile yet. Then you play, from your mini-hand of 3 cards, Let’s Keep Doing That. Nothing in your mini-hand is in the discard pile yet. You must pick your Action out of the discard pile right then, as part of your play of Let’s Keep Doing that, so, no, the last card from the D3P2 is not yet in the discard pile, available for use with Let’s Keep Doing that. It will be after you’re done playing both of the cards you choose to play, and not before.
Note that this ruling will also apply to Draw 2 and Use ‘Em (D2UE) and Fizzbin. Cards executed from your temporary hand are not in the discard pile until the whole Action is completed.
A: You can absolutely win. You just have to play the card that makes you win as the FIRST card. At that point, you just win, and you’re not required to play the other card you drew for Draw 2 & Use Em, since the game is now over.
The situation would be similar if your conundrum was in the middle of Draw 3 Play 2, though this issue is less likely to confuse people, since it’s pretty clear that they have (slightly) more options. Fizzbin also creates a similar situation.
…Since these cards are all played as part of a single play, would the player get to finish playing them out, or would their turn just stop? And if it just stopped, what would happen to any unplayed cards? We’ve been letting the player keep them.
A: The clear answer is that if you decide to play one of these turn-ending Actions first, of course your turn would end immediately, as you have been playing it. Of course, in the case of Fizzbin, you don’t get a choice as to the order of cards played, but that card most closely resembles D3P2/D2UE, since you set your main hand aside, and are working from a temporary hand of extra cards to execute the Fizzbin.
However, there is NO way that any remaining cards would go back into your set-aside hand. They are never intended to go into your actual hand at all, as indicated by the requirement to set your hand aside. Any cards left unplayed when you played the turn-ending card are discarded. If you wanted to play them, you should have done it before the turn-ending card.
Brain Transference: Discard remaining cards in your temporary hand and trade places with the player of your choice. Turn ends.
Time Portal: Choose a card as described and add to your set-aside hand. Discard remaining cards in your temporary hand. Turn ends.
What Do You Want: If you choose to take a Keeper or Goal out of the discard, it goes into your set aside hand. Discard remaining cards in your temporary hand. Turn ends.
I’ll Be In My Bunk: This card does not specifically say that your turn ends immediately, but you certainly can’t continue your turn if you “Excuse yourself from the game and leave the room for a few minutes.” Discard any cards remaining in your temporary hand. Leave the room.
Swap Plays For Draws and Get On With It, while they do involve having your turn end immediately, are New Rules, not Actions, so, as you can see from this answer, things would work a bit differently:
See: Can you Get On With It or Swap Plays For Draws with your cards from an Action like Draw 3 Play 2?