(Solutions)
In all cases, you need to turn the Iron Palace to give your attacker Toughness one on the first attack, so the Deathtrap doesn't kill your character. However, you need to wait until you seize the Eagle Mountain to play your Sinister Priest, because Tim can intercept and kill the character. The only exception is the Evil Princess because you need the extra damage. In most (but not all) cases you can substitute playing a Sinister Priest for playing Lateral Reincarnation to bring a Friends of the Dragon into play instead.
The Tortured Memories and Alabaster Palace are Red Herrings- you don't have enough power to make use of them.
Step 1: Play Netherworld Return, return a character.
Step 2: Turn the Iron Palace to give the character Toughness 1.
Step 3: Play LaGrange Four for 2 power.
Xiang Kai
Play another character. (1 power)
Attack the Eagle Mountain, do 2 damage to your Mountain Fortress, and Tim's character. Use Iron Palace. Hit for 6.
Attack The Palace with your Priest for 1 damage.
Attack, do 2 damage to your Mountain Fortress (or the Palace), and another character. Hit for 8 more, for a total of 9.
(You have to make sure you attack with the Priest separately from Xiang Kai, because otherwise you'll have to do 2 damage to one of the characters you want to do damage with. The effect isn't optional.
Xin Ji Yang, Evil Princess (5 fight)
Play a Sinister Priest (1 power)
Attack, do 1 damage to your Mountain Fortress and Tim's character.. Hit for 6.
Attack, join with Sinister priest. Do 1 damage to your Mountain Fortress and the Palace (or the Obsidian Mountain) Hit for 9.
(You need to play the Priest before you seize the Eagle Mountain, so you have to kill the Friends of the Dragon so it won't be able to damage the priest. Optionally, you can ping the Obsidian Mountain on the first attack, and the Friends on the second.
Flambards (4 fight)
Flambards do 1 to Obsidian Mountain and 2 to all other sites.
Attack the Eagle Mountain. (6).
Unturn and attack the Palace of Virtual Light (6 damage.)
Play Lateral Reincarnation. Flambards leave play, and do 2 more to the Palace. Bring a Friends of the Dragon back. Attack for one more point of damage.
(The trick here is triggering Flambards twice and still being able to take the Palace with combat damage. Lateral Reincarnation provides the answer.)
Walker of the Purple Twilight (6 fight)
Attack the Eagle Mountain for 6.
Do 2 damage to any target- either your Mountain Fortress, or the Palace.
Unturn and attack for 6/8 more (8 total.) Play a Sinister Priest and attack for 1.
(Pretty easy. There's even two ways to do it!)
Keeper of Hearts (6 fight)
Attack Eagle Mountain for 6
Attack Palace for 1 with Sinister Priest
Unturn Keeper, sacrifice Priest, attack for 8 more.
(The only trick here is getting 3 damage out of the Priest- 1 for the attack, 2 for the keeper)
Fire Martyr (2 fight)
Attack, Fire Martyr is now 6 fight.
Attack again, Fire Martyr is now 10 fight.
(Pretty darn easy- you don't even need to spend any power.)
Lord Shi (6 fight)
Attack for 7
Attack with Sinister Priests. Each gets +1 Fight. 9 damage.
(Trick is that the Priest needs to attack with Lord Shi, a counter part to some of the previous ones.)