1. Advice: Clear out the threat that's massing on your border. It probably realized that it can't get through and started parking more and more ships on the wormhole. Letting it spiral out of control is an easy mistake to make. A lightning or armored warhead works wonders and costs relatively little AIP.
2. Yes, the signals will be in your territory. I did the same thing, capturing planets first then starting the Fallen Spire campaign to cheese it out and not have to deal with AI spawning right next to the shard.
3. They can only spawn on planets that have a warp gate on the planet next to them. Say you have planets A-B-C in a line. You control A and the AI controls B and C. If you destroy the warp gate on B, they can't spawn waves on planet A. The best strategy is to destroy warp gates next to planets you really need and leave one planet with a warp gate next to it. That planet can be used as a "Fort" planet where all the AI waves will go. However, do not destroy all the warp gates connected to your planets. If you do that, then the AI will be able to attack anywhere. That would be bad.