First, welcome!
Second, I'm not electronically savvy at all so I can't help you. Sorry.
EMG's are active, aren't they? Thus their amplifier may act as a radio transmitter and something in your amps acts as a radio receiver. The reason why you can't hear your phone through the pickups is that the sound of your phone isn't made by vibrating ferric metal within the magnetic field of the pickup - a microphone that picks acoustic sound works differently. Speaking of amps picking radio signals, back in the day I made all sorts of wire loops from the amp to a pair of cassette decks to record multiple takes on top of each other. Often that mess also caught a Russian AM station!
Why you don't get sound when fully plugged in... Start from the easiest part, the jack. Triple check that the ground really goes to the sleeve and the hot to the tip. Also check that when fully plugged in the hot prong doesn't touch anything grounded, including any copper tape or shielding paint inside the cavity. To test that simply unscrew the jack and pull it out of the guitar, then plug in and see if it works.