OK, now to answer the questions.
Assuming that your house is wired in the same way as most other US houses, the phone sockets only have two wires connected in them.
They may have 4 but only two carry signals.
That said, there should be no reason why the phone company would be visiting your house to move or change anything.
In my neck of the woods, if you sign up for DSL, they send you the modem and the filters in a box and tell you to plug it in.
With the computer in the original location WITHOUT the router connected, find the IP address. You can do this by going to Start, Run, type CMD and press enter.
Now type ipconfig in the black window that appeared.
This will show you the IP address of your computer. If it starts with 192.168 or 10.xxx then you do not need a router as you already have one.
Let's start there