How does the position of the center of pressure change with the use of different high lift devices?
When I use a high lift device that just increases the chord length, it moves backwards so that it stays at the same percentage of the length of the chord


Only when the Fowler flap does not change the airfoil's camber, which is rarely the case. Normally, the flap will not only move backwards, but also rotate trailing edge down. A bit only for the take-off setting, but a lot for the landing position. This camber change will move the center of pressure back even if there were no surface increase, and the extension of the wing at the trailing edge will move the center of pressure further back again.


  When I use a high lift device that moves downwards, it has the same affect as increasing the AoA, so the center of pressure moves forward


Again, no. The camber change dominates the position of the center of pressure, and it will move back since the wing now sports more rear loading (more lift is created at the rear end of the airfoil due to local camber increase).