OpenReach look after the cables and boxes up to the master socket, after that it is yours, but you can't deal with OpenReach the telephone or internet provider has to do that, your problem it seems is as yet you have no master socket, in the old days telephones did not plug in, they were hard wired, and the GPO as it was then looked after everything, including the phone.
As to DIY well it is about 50 volt, so no big shocks, and it comes into the house as two wires, internet and cordless phones only need two wires, but older phones need to have a capacitor and resistor network to ring the phones bell, and the two wires much be right way around, unlike internet and cordless which don't seem to care.
What you should do is get your telecom provider to fit a master socket, this has a part which can be removed by the user, and internal hard wired phones connected to it, so if you have a fault you can remove this bit of the socket and test phone with all internal wiring disconnected, so you know if OpenReach fault or you wiring at fault.
Pre-internet and broad band I would have not worried about wiring my own extensions, but the phone wires are twisted together, and the rate of twist is important to get full internet speed, so DIY wiring works A1 for the phone, but internet and fax it needs more care, not that we see much of the 19th century fax any more, it is still used by some people as caller ID means you can know who has sent it, more secure than email. But not as good as teletext.
Today it seems we are moving to fibre, and the hard wired telephone is for the museum, the big problem is a power cut can mean no phone, even having a battery back up in the home does not help, as need power to boxes in the street.
Many around here are giving up with land lines, seems cheaper not to have one, but it depends on your skill as to if you can DIY, personally I now have one telephone socket, and all cordless phones, and router next to socket, in the past I had wires all over the house to work the fax and telephones, but things have moved on, no need today to listen to see if neighbour on the phone, and if not press the party line button.
I must be getting old, I have used the phones with handles you had to turn to make them ring, and number of times you turned it told people who you wanted. And high winds they stopped working.