Can I install GSX in my "new" computer without problems, using mu own GSX serial?
Yes, of course. Your license will always be valid, you have 6 activations per order, and they rotate every 90 days, and you can also deactivate a system so you won't even consume a new activation on the next reactivation. It's almost impossible, under normal use, to be out of reactivations and, if that really happens, you can always email support for a manual reactivation.
Your GSX license is valid for FSX, FSX:SE, P3D 1, 2, 3 and 4. Considering you are migrating to P3D, we strongly suggest P3D V4.