What conditions must be met for the Legionnaire to dock? For example, does the target ship have to be immobilized and its shields overcome, or can the Legionnaire dock mid-flight?

The conditions are the same as current ship-to-ship docking as featured on the Constellation and Merlin. So, the ships must be aligned and orientated correctly. This can be done at any speed, although the slower the better. The only difference is the Legionnaire can override the docking permission system usually controlled by the target ship.