1: If the Active player activates a system he controles with a PDS in it, can that PDS fire on a enemy ship outside the Activated System (assuming the Active player of course has the Deep Space Cannon tech).
No. The only PDS fire that can occur is in the system activated by a tactical or transfer action--not adjacent systems.
2: If the Active player activates a system he controls can all players with Adjacent Systems with PDS and Deep Space Cannon tech fire upon the Active players ships inside the Activated system after movement?
Yes. And any enemy PDS in the activated system can fire regardless of Deep Space Cannons.
3: If the Active player activates a system that's adjacent to 2 different systems of different players that have PDSs with Deep Space Cannon tech can both players fire their PDSs on ships of both the defending and attacking player in the Activated system after movement is completed or only on the Active player?
No. The target of all third party PDS's is always the active player. If you active a system, all enemy PDS fire will be directed at you.
4: If the Active player activates a System he does not control and the Active player has a PDS in an adjacent System with Deep Space Cannon tech. Can that PSD only fire into the activated System or can it also fire on a different adjacent System? (because if question 1 is a yes, then this should be possible to, right?)
The PDS can only fire into the system activated. If you want to fire a PDS into a system, you (or your opponents) have to activate that system.
5: If the Active player activates a System he does not control and moves his ships into the System. Can other players who are in adjacent Systems to the Activated System fire on ships of both players or only on ships of the Active player?
PDS's belonging to third parties always fire at the active player.