Does it work to give you any 3 green you want, as long as you pick a 3 movement?

No. There's errata listed in the FAQ for Stay On Target, but the upshot is that if you use it, whatever maneuver you end up executing this round is red.

Yes, however Stay On Target tells you to treat it as a red maneuver (it doesn't actually change it.... and this interaction is probably why) So it would be a green maneuver that doesn't clear a stress and gives you a stress, which you can't do if already stressed.

Yes, however No, Stay On Target tells you to treat it as a red maneuver (it doesn't actually change it.... and this interaction is probably why) So it would be a green maneuver that doesn't clear a stress and gives you a stress, which you can't do if already stressed red maneuver.

Also, you do realise that Unhinged Astromech also says to "treat them as green maneuvers", right?

As for the interaction between Stay on Target and Unhinged...

There's a FAQ entry that states that if 2 effects change the difficulty (color) of a maneuver, the effect that makes the difficulty harder trumps the other one. Therefore, Stay on Target makes the maneuver red, even if you have Unhinged astromech, R2 astromech or Nien Nunb on your ship.

The move will be red the worst effect always wins.