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Does a Skimmer that used a Scout move to move more than 12 inches get a cover save if it's opponent Siezes the initiative?

No, because for skimmers, to get the cover save they must have moved flat out in the previous movement phase. The scout move is NOT a movement phase.

A Bike that Turboboosts does however, as they get the save in the Following Shooting phase, regardless of when they turboboosted.

No. Scout moves do not occur in the movement phase. BRB p71, "Shooting At Skimmers".