the banjo nut is the thing that holds the frontblock onto the body. In this case it has been replaced with an lpc which is basically an extended banjo nut that is hollow to allow for more air volume.
the jam nut is the thing that goes around the valve in the lowertube and the valve retaining screw is the thing that goes up into the valve from under the body.
the trigger sticking could be because of several reasons. Paint and or gunk could have gotten into the frame, the 3-way may be dirty or the timing rod may be bent causing binding. Start by taking it apart and cleaning everything
use wet/dry sandpaper and water for polishing.
I have videos over disassemble, cleaning, polishing, etc for autocockers on my youtube channel https://www.youtube.com/user/halfsquat41?feature=mhee