Can you tow a tangled derelict out of a sargasso sea?
No, it would have to be repaired first.
You could also use one of the abilities that move other ships, if it’s pushed completely clear of the terrain.
Can a ship shoot after freeing itself from a sargasso sea? (by extension, is “its action for the turn” a move action?)
Not on the same turn, unless it had an extra action available. It uses the ship’s standard action for the turn, but is not specifically a move action.
Does Parley last for the duration of the attacker’s turn or until the owner of Parley starts their turn? (or a whole round)
It’s only for the duration of the current turn, not the round. In a multiplayer game, the ship with Parley would have to use it on each opposing player individually to avoid being attacked.
Does a roll of 5 with the Fear keyword affect abilities for just the Fear owner’s turn, or the whole round?
The 5 result applies only to the current player’s turn, so it’ll shut defensive abilities and continuously active effects like point reducers, but won’t do anything to own-turn abilities like a captain, a helmsman, or an ability generator.