First off, apologies if I'm repeating past questions. Tried searching, but was not so successful. If something's been answered before, feel free to just link me to the respective thread.
1: First game, we ran into an interesting scenario. The rewards of the final adventure completed included one doom token and an elder sign. The elder sign would lock away the ancient one, the doom token would have released him. Which resolves first (I would assume it's the player's choice)?
2a: It would seem that you are supposed to declare the use of a spell before you roll the dice, implying you must simply anticipate needing to use it, is that correct (I think the answer is yes, you must declare it)?
2b: If so, what if you only use one of two slots on the spell (for a two-slot spell), can you fill the other one later on another adventure, or later on the same one if you fail the adventure on your first attempt?
2c: Do you have to keep the dice locked up on the spell until they are used, or can you elect to simply "dispell" the dice to free them up for another player?
2d: What if you end up not needing to put any dice on the spell (in other words, you roll well and succeed at the adventure without needing dice reserved)? Does it go away, or remain in play until you place dice on it? Or is it not a choice - you have to put something on it - in which case what if you used all the dice?
3: If an adventure has a task that requires loss of stamina or sanity, does that happen regardless of if you complete the overall adventure (I think the answer is yes)?
4: Do the yellow and red dice last only for the single roll immediately following the use of the item that brought it in, or for the entire turn trying to complete the tasks?
I would suggest you also have a look at the BoardGameGeek Elder Sign forum as that tends to be more active than this forum. In particular, threads like From the Arkham Nights Event ~ Many Questions Answered
1) The consensus seems to be that such Rewards and Penalties are applied in left to right fashion as printed on the Adventure card.
2a) While the rules as written so state that you need to spend the Spell before the dice roll, a lot of people now play that Spells are spent after the dice roll. The main reasons are (i) the designers have indicated that is the way they play and (ii) Elder Sign: Omens play spells this way.
I prefer playing Spells after the dice roll as it removes a lot of issues. It does make it easier but I prefer easier over frustration.
2b) You must fill both dice slots if you can. You may not fill the other slot at a later time if you can fill it now.
2c) Either dice are "locked" until they are used. There is the question if you must use a locked spell dice if you are able, I think you must. I don't see why you should be allowed to "dispel" a dice if it is not going to be used.
2d) That's why I play that Spells are played after the dice roll Either way, I think you must play (i.e. "lock") dice on a Spell if able.
3) Yes. You take the gamble that you may not complete the Adventure in which case the Stamina / Sanity loss has been wasted. Note that the situation may be considered different if it is a Monster Task with Stamina / Sanity loss. Even if you fail the Adventure you still gain the Monster task as a trophy, therefore it isn't really a loss if you take the Stamina / Sanity loss,
4) The Yellow / Red dice are added to your dice pool. They remain in the dice pool until you use them, so they remain with you for your entire turn.
Here's what I said in another BGG thread
The Yellow / Red dice are added to your dice pool
When you complete a task you continue to roll the remaining dice from your dice pool to see if you can complete another task.
When you fail to complete a task you set aside one die from your dice pool and roll the remaining dice to see if you can complete a task.
So, the Yellow / Red dice stay with you for your turn (unless you set one aside after failing a task or the dice gets locked for some reason).