have you tried connecting to the test socket if still not working there it is not chargable
If the test socket is dead, then the fault is outside, and the £130 charge will not apply. All you can do is to request another visit.
You also mentioned a junction box, is that before the master socket, as damage to that, if its inside your house, may be chargeable?
Provided that the cables on the outside wall have not been damaged, as described on that page, then you should not be charged.
Make sure there are no other sockets that may be working, as the line test does not show a fault, this could also be due to a wrong connection in the street cabinet, and the line is going to a different customer.
Its actually Openreach, not BT Retail, that look after the external network.
Its difficult to say, its going to depend on what the Openreach Field Technician says when they close the task, after the visit.
If there is only one socket, then they may remove that box.
The Home Improvement Charge is just that. If the wiring needs updating, then that is what they will do, and you will be charged.
In the long term, updating the wiring is going to reduce the chances of a fault within your house.