Beginner lifters' first priority needs to be making sure their posture and body positions are correct (relative to the barbell). All beginner programs need to include different exercises that develop these proper positions. That being said, I wouldn't recommend doing ONLY Snatches and Clean & Jerks to become a better weightlifter.
So to answer your first question, I would start on a weightlifting program. You can still work in bench press, all weightlifting programs include Back/Front Squats and I suggest doing Clean/Snatch DLs instead of the traditional DLs or Sumo DLs.
Stick to one programs periodization method. In other words, don't try to continue your current strength program with another weightlifting program.
There are several programs on this site that have been proven successful for a lot of people. You can check them out here: http://www.catalystathletics.com/workouts/cycles.php