I am use to the free movement of the external camera on X-Plane 10. Whenever you're flying and looking at the plane externally, you can freely rotate your mouse and look around all axises. But in FSX/P3D whenever you do this, it only rotates in a circle around the plane in one axis. How can I make this camera behave exactly like X-Plane?

P3D does allow you to have outside views from all angles as FSX always did.  The camera view "Outside View" gives you this ability.

You might want to look into EZ Dok.

Holding down Spacebar and moving the mouse gives you full movement around a sphere centred on the aircraft, always looking in at the aircraft. Shift+O to toggle on or off if you don't want to hold spacebar. If you're talking about looking away from the aircraft, no, you can't do that on Spot views, you'd have to use an external aircraft camera.

Richbarry states it well. The only thing I'll add is that if you assign the view directions to a single hatswitch, you'll be able to one-stop-shop. Having played with the X-Plane recently, I found it to do a bunch of things well (omg let's not even talk about night flying) but the view modes--if assigned similarly--work much the same. At least the way I remember it...