PD activates as soon as I enter the defence radius, doesn't matter if I get close to planets or not.
Nope, they have to be in sensor range, hence part of the description which says "Can accept targeting information from any fleet or planet in it's area of effectiveness." Basically saying "if you can't see it, I can't hit it".
This one is difficult. Big fleets repair damage between engagments, so they can face off against smaller fleets and repair between each battle.
Again, I think you're wrong. They only "repair" because instead of putting up that ship that had been hit last round, they put up a new ship that hasn't been hit yet. This makes it look like you're repairing ships, but fight the fleet enough and eventually they have to put in a damaged ship. I think if you have a turn or two between battles they repair (remove the yellow ring) - or it may have to land on a planet.
