Fish low voltage cables through the plumbing chase

If you don't run conduit, you won't be able to pull wires with a pull string going through holes in studs anyway. So you might as well firestop them (or run conduit and firestop around the outside of it.)

If this were my house, I'd figure out where those cables are on the other side of that wall, and pull them back into the house, patch the holes, and arrange connections inside.

• Your telephone runs are going to be terminated together, for whatever landline service you get. The telephone provider should provide a Network connection jack.
• Your TV lines will get connected to your service provider's equipment (when you get one) or antenna/amplifier/splitter.
• Your Ethernet drops will get connected to a switch or router connected to your Internet providers equipment. (perhaps that cell box you mention)

The only outside gear should be that provided by the service providers, and they will bring wires inside the house to your connections.

The electrical codes really don't say much about low voltage wiring, which this all is. You can run wires in walls, provide connection plates, etc. Do it safely and sensibly.

To figure out what goes where, look into signal tracers as mentioned in other comments.