Off to a good start. In 2nd part of the video your knees were more bent, keep working on that. Try to keep your shoulders in line with the board. A good drill to do is to ride with hands together, first in front, then in back and practice bending knees.
Totally agree with Lamps here.
1. Align your upper body with your board as much as possible. Holding your hands in front or back is an excellent tactic to improve shoulder alignment. Another tactic is to hold on to the bottom of your jacket at your sides while riding, for the same effect.
2. Yes, flex your knees. As Poutanen points out, it will improve your ability to adapt to changing terrain. The run you were on looked like a very smooth groomer. When you get into chop/crud, a static body position will have you bouncing around and falling.
3. It looks like you're ready for steeper terrain. This run looked like a walk in the park on a snowboard. The tips we have all mentioned will become evident on more challenging slopes.
Keep up the good work!