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.

Source: https://spacenews.com/spacex-plans-to-start-offering-starlin...

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.

Perhaps not initially, but IIRC, they use "digital" antenna so it should be upgradable to track in real-time. Perhaps even by an OTA update.