If you some how managed to get your "in" and "out" ports mixed up, your gauge would read 1200 psi all the time and you would not be able to adjust it. Just something to check on.

To answer your questions:
1) no, its not normal. There are a few reasons this could be happening. We need to know if you are using HPA or Co2.
2)No, you do not have to adjust both. Usually we use the Stabilizer to get the velocity in the general area of where we want it, then use the FVA to make the fine tuning.
3) Adjusting the Stabilizer will adjust you velocity. When you lower the pressure the stabilizer is set to your marker will shoot slower. When you raise it back up, your speed will increase.

The two reasons for using a Stabilizer are to control your air/gas source better so that your marker it more consistent and you also are usually able to lower PSI entering the marker so that not as much air/gas is wasted. Just having the Stabilizer on there will make it more consistent, not matter what setting or level it is at. So, the ideal way is to lower the Stabilizer setting down low enough to save as much gas/air but still allow the marker to operate. Go too low and the cyclone will stop working and the maker will not recock automatically. Find the a level (usually around 600-700 psi) and set it there. Then adjust with the normal velocity adjustment screw to fine tune it. If you can't get it up to speed, you will have to raise your Stabilizer setting some until you can.

__________________