(1) Does this action target?
No, because the ability affects “all dice”.
(2) Does this mean that any die in either player's field that has more than one copy of it will be affected?
No, the ability affects each player’s Field Zone independently. For example, you initiate House of Mystery and you have 1 Wonder Woman die and your opponent has 1 Wonder Woman die, neither player would be affected (i.e. you couldn’t move your opponent’s Wonder Woman die to their Used Pile). However, if your opponent has 2 Wonder Woman dice, then one of those dice would be moved.
(3) Is every die that has more than one copy of it affected, or does the effect only move the copies/multiples of a character to the owner's used? (like... one die remains after the duplicates are moved)
Only identical dice are affected, one die remains. For example, if you have 3 Sidekicks and 2 Mister Mind dice in your Field Zone, 2 Sidekicks and 1 Mister Mind die would move to your Used Pile, leaving 1 Sidekick and 1 Mister Mind die remaining.
(4) Does this affect both players?
Yes, but within the parameters stated in question 2.
(5) Are dice that are "copying" another die affected?
Dice with the ability stating that die “
becomes a copy” of another die (e.g. Doppelganger, Mimic, Morph, Mr. Mxyzptlk or Taskmaster) would be affected by House of Mystery.
(6) Are Sidekicks affected by this action?
Yes, because they are Sidekick Character dice when in the Field Zone. It also affects Action dice and tokens, since the ability does not specify a die type.