Cutting the mirror: Glazier.
Motor: Well the picture doesn't really show enough, but I would guess that it is a general one.
You will probably have to get the motor/mirror thing going and then fire your laser at it and see what happens.
If it "fails" as in the "image" - which I am guessing is what will be made - is wrong, you may have to adjust the speed of the motor, or the timing of the laser being fired to sync them.
This could be done by having a detector system on the mirror and when it passes the sensor, it tells the arduino that the mirror is at "point x" and so they both sync that way.
Good luck with it.