Does Q1 or QW have it in any form...
I mean any engine or OS.

Not that I know of. I think the problem is that projectiles don't make any sound and so the doppler effect wouldn't be very noticeable. Ideally you would need to add a thruster sound to flying rockets to make the doppler effect worthwhile.

Sound could be stealed from q3. No problemo comrades.

I had rockets emit QW flame noise from /id1/pak once upon a time in JawnMode, it didn't sound too bad. Would save having to use a 3rd party sound...

Player can move with quite big spin. There could be doppler effect for any ambient sounds, shots.
Funny but useless.

Doppler would only work like that if QW considered the player to have 2 separated ears (it doesn't)