I have a four wire phone line coming into my house. I can't for the life of me figure out who owns it. It's not Verizon, not wire star, who else could it be? It's Coastal Dr. 77845.

You own it. At divestiture everyone restructured and made inside wiring the building owners problem. I know several people that owned computer companies on that street when the area was developed and they may have had a dedicated circuit. The circuit ID information is likely in the interface box on the back wall of the house. It also typically is written on the can that the NIU is located in.
Don't expect Verizon to claim to own it and for sure not to come out and remove it.
Pictures or Brand/model info might give us information on what type of circuit it was.

The thing is, I don't want it removed. I wanted to see if I could get deal service from it!

So you want to know who provides land-line service for DSL to the junction box? That would be Verizon. You own everything that connects to the box downstream, but not the box.

That's what I thought, but I called Verizon, and they said they cannot provide service to that address.

It should still be Verizon/Frontier's line. Doesn't mean that you can get DSL over it. If you are too far from one of their DLSAMs, DSL internet is unavailable at your address.

If Verizon is telling you they don't service your address, call back and speak to a different representative. Had this happen in our old home in edelweiss gartens (we had moved from sw valley and had service) but the representative was telling me that they didn't service our area even though I continued to tell her we had service at our other house that was literally 2 miles away in the same city. Hung up from her, called back, got a different rep and signed up for service with that rep.