We already have satellite communications for media and voice, but iirc they have fairly limited bandwidth constraints. I think figuring out a way to get the flight recorder data to operators in realtime would be the first problem to solve, and I don't think it's been done yet for commercial flights.
There was a startup around like 5-6 years ago (gone now, can't find any info on them, their name was "Flamingo" something or other) that had a proof of concept solution but they burned out quickly.
Yes, it can, although I can't be sure it will work at 500 mph. They tested it with a C-12, which has a top speed of 338 mph, and were able to transfer data at 610 Mbps.
Yes. The USAF has already tested it successfully on several aircraft.
A Starlink receiver always has the satellite(s) in fairly fast motion, and phased array antennas can "move" quite quickly, so it's possible that there's no particular challenge to an aircraft scenario. Then again, the basic rooftop unit may make some simplifying assumptions and aircraft would require better hardware and/or software, but that's not a big problem as they can handle a higher price point.