When you move fresh Loyal ships, you may move any of them one more time. (Damaged ships can move with them once, but won't move one more time. Resolve this after all Catapult moves.)
A: No, it is not recursive. The more verbose but correct text is: When you move fresh Loyal ships but not with Sprinter Drives, you may move any of them one more time.
A: You can Catapult move with the bonus move if you're moving from a starport.
A: Choose a group of any number of fresh Loyal ships that just moved, and move them one more time. So if you dropped off two ships in the same gate, you could move those two ships again.
A: No, only ships that are both fresh and Loyal, and that just moved, may take the second move.