A few radio technical comments
I've been reading the thread, and have a few comments based on my radio/RF Engineering/software/comms background (my aviation is only P-3 aircrew, private pilot, and CAP SAR):
"Ping" - we don't know if this is being used as a technical term or simply a shorthand way to describe a transmission used just for link establishment/maintenance. It might actually refer to an ICMP "ping" message, but I doubt it. The safest assumption is the most general - it's just a received, short transmission.
Regarding the search arcs - they appear to be at constant range (and elevation) from one satellite. This implies that they were established either by signal strength measurements or timing. Triangulation, and measurements with two satellites don't match this.
Without knowing deep details, we cannot be sure of which. I would guess they are using just signal strength. The satellite probably logs each message with a bit of RF data - frequency/channel, strength, antenna used. In either case, unless remarkably tight timing information is being kept, the arc position will not be very accurate. If signal strength, they probably used one or more pings when the position of the plane was known to establish a baseline.
I hope someone with deep knowledge of INMARSAT appears and comments.
A non-technical note: the arcs appear to correspond to just one ping - probably the last. We have not heard where the other pings were located - unless they, by some chance, just happened to also be on the arc (i.e. had the same signal strength). A question to be answered.
Another non-technical: I doubt the aircraft had to be flying to generate the pings - it just had to not have been destroyed or completely powered off.
Regarding cell phones at altitude. Radio signals at those frequencies (low noise) can travel a surprising distance. A ~1/2 watt cell phone can easily reach 100 miles, unless TDMA timing protocols rule it out (depends on the specific modulation scheme). Likewise, doppler from a moving aircraft is from the component of motion along the line to the tower. Thus, if the phone is talking to something at 90 degrees to the line of flight, there is zero doppler, with it increasing as the angle approaches 0/180. I recently had an email appear while I was riding at >FL300 and had forgotten to set the phone to airplane mode.
The same observation on radio signals at high frequencies means that hand-held walkie talkies could be used at quite a distance for communications from an aircraft. Even an FRS radio (cheap HT's sold at many stores) could be used.