I cannot set the speedbrakes / spoilers to automatic, so they deploy upon landing. When I try to set them, they actually deploy a bit. What is wrong?

Nothing is wrong. The real 727 does not have the ability to have the spoilers automatically deploy upon landing. You must acitvate them manually once you land. There is no notch for the speed brake handle for the auto position. This is where the 727 has something in common with the 707, and not the 737. ;-)