Say, for Cruel Edict I have to choose which one of my opponent's heads sacrifices a creature. But I think that Sudden Disappearance makes all of their permanents go away, not just one person's. Can someone tell me why?
Cruel Edict and Sudden Disappearance both only affect a single opponent. In general, the only cards that would affect both heads of an opponent in two headed giant (2HG) are cards that say each or all within the rules text (example: Heartless Hidetsugu), but this would be have the same effect in non-2HG duels or multiplayer games.
The only differences in 2HG with respect to rules text, is that players attack/defend in teams and take their turn as a team. A card that referred to an attacking or defending player would be referring to both members of the teams, or cards that add/skip turns or phases. The MtG Comprehensive Rules state:
810.7a Each team’s creatures attack the other team as a group. During the combat phase, the active team is the attacking team and each player on the active team is an attacking player. Likewise, the nonactive team is the defending team and each player on the nonactive team is a defending player.
Example: One player in a Two-Headed Giant game controls Teferi’s Moat, which says “As Teferi’s Moat enters the battlefield, choose a color.” and “Creatures of the chosen color without flying can’t attack you.” Creatures of the chosen color without flying can’t attack that player’s team.
Example: If an attacking creature has forestwalk and either player on the defending team controls a Forest, the creature can’t be blocked.
805.8 If an effect gives a player an extra turn or adds a phase or step to that player’s turn, that player’s team takes the extra turn, phase, or step. If an effect causes a player to skip a step, phase, or turn, that player’s team does so. If a single effect causes more than one player on the same team to add or skip the same step, phase, or turn, that team adds or skips only that step, phase, or turn. If an effect causes a player to control another player, the controller of that effect controls the affected player’s team.
You are also "sharing" turns.
A card that reads in part "target player takes an extra turn" will cause both heads to take an additional turn.
The same is true for controling another player's Turn (Mindslaver, Sorin Markov)
Sudden Disappearance affects only one opponent, even in Two-Headed Giant.
Players do not share control of permanents in Two-Headed Giant.
810.5. With the exception of life total and poison counters, a team’s resources (cards in hand, mana, and so on) are not shared in the Two-Headed Giant variant. Teammates may review each other’s hands and discuss strategies at any time. Teammates can’t manipulate each other’s cards or permanents.
Cards that refer to one opponent only affect one opponent. Cards that refer to multiple opponents affect both. The only change to what you'd expect from normal Magic is for certain cards that say "attacking player" or "defending player":
810.7b Any one-shot effect that refers to the “defending player” refers to one specific defending player, not to both of the defending players. The controller of the effect chooses which one the spell or ability refers to at the time the effect is applied. The same is true for any one-shot effect that refers to the “attacking player.”
Any characteristic-defining ability that refers to the “defending player” refers to one specific defending player, not to both of the defending players. The controller of the object with the characteristic-defining ability chooses which one the ability refers to at the time the nonactive players become defending players.
All other cases in which the “defending player” is referred to actually refer to both defending players. If the reference involves a positive comparison (such as asking whether the defending player controls an Island) or a relative comparison (such as asking whether you control more creatures than the defending player), it gets only one answer. This answer is “yes” if either defending player in the comparison would return a “yes” answer if compared individually. If the reference involves a negative comparison (such as asking whether the defending player controls no black permanents), it also gets only one answer. This answer is “yes” if performing the analogous positive comparison would return a “no” answer. The same is true for all other cases that refer to the “attacking player.”