It applies only to a move action that begins within L of the ability source. If the sea monster gains a second action but is no longer within range, it does not get the bonus for the second action.
How does the ability of Intermediario interact with Mysterious Islands? Would his ability bypass any effects of revealed MI’s? As in, I wouldn’t need to roll for effects if I use Intermediario to get treasure onto a nearby friendly ship(s)?
They don’t interact at all, because Intermediario’s ability doesn’t involve docking. It will not reveal a face down MI, and it will ignore any ability of a face up one (no roll required or allowed).
If you do dock the sea monster at a MI before using the ability, you have to completely resolve the island (reveal, roll, apply instant effects, etc.) before moving treasure to other ships.