I picked up a copy of this game just over a year ago and came across many of the same rules issues that you seem to have. I will list the conclusions that we came to after a few games, hopefully they might help you out.
1) When the Dark Legion players draws an event card and it states that some effect starts in the next round is that:
a) From Dark Legion player's turn, to Dark Legion player's turn.
b) Just the next player's turn
c) The next round of turns for every player starting when all five of their team tiles are placed face down again, and the first team is drawn for the round?
We play it as (a). We found that leaving the card face up next to the Dark Legion player until the next DL turn is the easiest way to remember the duration of effect. With the way that the turn initiative is determined this can lead to some teams being subject to the effect twice. While this can seem a bit unfair it really isnt that different to a player getting two turns before another gets his turn anyway so we dont worry about it. Anything in the interests of keeping the game flowing fast.
2) When exactly are you allowed to play Doomtrooper cards?
b) Your turn only
c) Your turn only, and it counts as an action
We play (a). Trying to work out rules for timing proved too fiddly so we went with the simple option. It has worked..so far.
3) If there is a doorway that is only one space wide, and a figure is on either of the single spaces leading to, or from the room for this doorway, can you move between the figure and wall?
Can figure "2" move to any of the spaces marked "A" if "----" represents walls?
Rulebook p.6 under "MOVE:"
"You may not move through another figure or through walls, but you may move diagonally between two figures"
I dont know how other people deal with this generally but we tend to call this issue "blocked diagonals" in my group. Every miniatures game that uses squares to measure needs to address this somehow. Doom and Heroclix for example do not have "blocked diagonals" and allow movement without restriction as long as it isnt through an enemy figures actual square. Space Hulk on the other hand does not allow diagonal movement when either terrain or figures occupy diagonally adjacent squares (SH has "blocked diagonals").
In SotC we allow movement to any square that it not occupied by an enemy model regardless of whatever occupies any adjacent squares. Obviously this does not allow figures to pass through walls but it does allow them to move diagonally past the end of a wall.
4) If there are no entrances marked on the mission briefing for the Dark Legion players reinforcements does that mean they can just choose any entrance from a board that is considered in the Citadel? Or do they not get to play their reinforcements that are drawn from the Event Cards?
Like MrSkeletor said, any of the four boards that the cardboard arches sit on are considered "the citadel" and reinforcements can always enter onto those boards (entry areas are noted with the dark arrows in the first scenario but not in the others, which is confusing).
5) Does the Dark Legion player get a promotion point for every wound he inflicts, or only at the end of the mission is the total number of wounds currently on the the Doomtroopers added up, and then given to the Dark Legion player? (This is affected by Medic cards, and the autoinjector I believe
In the interests of simplicity we allow all relevant cards to be played at the end of the game and then award the final values to the DL player as per the final Doomtrooper wound values.
As the DL player doesnt have a pegboard to keep track of wounds inflicted as he goes we find it easiest that way and it is in keeping with the idea of that annoying card that allows a Doomtrooper to steal his opponents XPs at the end of the game I think.
Keeping track of the damage caused by the DL player immediately when it is inflicted is just as valid a method but we found the above method more streamlined.
I hope that some of that helps.