Think you have to configure the sticks on Windows. Don't think Wine can do this. That's hearsay though so YMMV. Mine are software anyway or just work™.
TrackIR API works. I demo this here: https://www.youtube.com/watch?v=D7n4hwABJFU (it's also on PeerTube but our instance is bust atm)
TrackIR hardware does not. That's a blackbox since v5. Tobii hardware does AFAIK not work but this may change any day. Some people started hacking at this again and again.
Also do check out the StarCitizen LUG and the helper scripts: https://github.com/starcitizen-lug/lug-helper
Both. It's complicated. Wild mix of my trusty X52 Pro, my own virtual joystick written in Rust and AntiMicroX and/or Moltengamepad.