1. So how is that done?
You can either push the button and then push or pull the control or turn the control to change the mixture.
2.Which direction for rich/lean?
Push or clockwise for Rich, Pull or counter-clockwise for lean
3.When to turn that thing (only when all the way in or out, or anytime)?
Personally I'd just push or pull it, you only use Full Rich or Idle Cutoff and it isn't intended to be used for altitude leaning.
4.Who is supposed to turn the control?
The PIC
5.According to what behaviour/indicator/parameter ought it be set?
The checklist/POH
6.How can I check if the setting is off?
If I was concerned that the setting was different to that commanded by the control, then I'd get engineering to have a look at it double-lively.