Hello! I'm a telecommunications architect from Australia. I don't work in the aviation industry, so I will endeavor to only discuss components pertinent to my expertise. If anything that I say is incorrect, please notify me so that I can update my correspondence to reflect this.
Both TelcoAG and snowfalcon2 appear to be on the money. Specifically, snowfalcon2 posted:
3. A third method may be to utilize the "Time Alignment" procedure that is widely used in e.g. GSM. [snip]
The caveat with this method, hovever, is that it is normally used only when you actually establish a connection to send a payload. For a periodic "ping" you normally don't need to allocate a timeslot.
By my own admission, I am not familiar with the TDMA signalling used on the Inmarsat-C communications platform, but it is my understanding that the signalling channels are also TDMA, meaning that it is highly likely that delta-T between timeslot allocations for handshaking as received by the two birds is how the speculative great circles have been generated.
4. It is quite conceivable that the Inmarsat satellite can measure the "elevation angle" of the received signal directly. Newer satellites have narrow individual "spot beams" to increase capacity, and obviously the system can keep track of which beam is allocated to any particular transmission.
Spot beams tend to be focused over areas where capacity constraint becomes an issue. Shaped beams (covering greater surface area) are used over sparely populated areas (such as oceans) because of the lower capacity requirements in these areas and the lack of business case for cost recovery in investing in additional hardware and power requirements on the space platform to accommodate underutilized transponders.
There has been a lot of speculation in the thread regarding the use of the SATCOM on the AC to communicate with ground crew in a hypothetical scenario where the person or people who have intervened in the operation of the aircraft intended to land it. Whilst it is true that the airframe of an aircraft is a decent Faraday cage, a pair of 5W VHF radios with a small antenna near the window of the flight deck would be sufficient to establish voice communications with a ground crew.