I don't know what you do up your end of the world, I was under the impression that ARINC ran the LDOC service and you called them up and then they patched you through on a phone line to the requested airline office.
Yes, that's what I used to hear. The pilot was instructed to stand by while a phone patch was set up. Sometimes that involved a change to a frequency which was unfavorable at my end, causing me to lose them altogether.
I remember overhearing a heated argument between airline maintenance staff and the pilot, during which the pilot exclaimed, "I'm the one who has to fly this aircraft!" In one exchange I monitored, a plane had its avionics door come open after takeoff, forcing the pilot to return to the airport in Vancouver, but that was on VHF when I lived there.
Medical emergencies were handled through a network named Medlink, I believe.