I don't think the Doppler shift resulting from 130 knots airspeed is going to be significant enough, considering that the satellites themselves travel at 15.000 knots, give or take
It is, considering that you're talking about centimetre-band frequencies, so unless the unit was specifically designed with high-speed applications in mind, it will in all likelihood never find the signal as it will be outside its scan range.
Tested the sat phone in a car. It is completely useless while moving. A brief obstruction (a few houses) blocking the line of sight and the connection is lost.
That is correct. I have been using Thuraya quite extensively for the last few years and yes, you can't use a handheld from inside a moving vehicle (but at least you don't get done for talking while driving
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Data transfer works fine most of the time if you're happy with 9600. When I first started using the internet it was a 4800 baud telnet job and I thought that was plenty. 9600 bauds while stuck in the back of beyond is the equivalent of fiber optic in London (in fact, it's much better). There is also the SMS feature, which is reliable and at a very reasonable cost (it used to be free up to a certain number of messages).
Cost-wise, as has been said, it compares favorably with using a GSM in roaming mode in many places. Thuraya to Thuraya used to be US$40/min which is cheaper than many standard GSM rates.
So, while most probably not usable in flight (I'm talking about a bare handheld), it is very cost-effective and a pretty reliable solution if travelling within its area of coverage. Certainly much more so than the SW radios we used to use before them (and still carry around, just in case).