When you jab, you are stepping with your left foot and leaving your right foot behind. Push off your right foot and jab simultaneously with the movement of the left foot; in the video you are stepping, then jabbing. That "step" should be more of a glide and when the left foot settles, the jab should be fully extended. Its a fencing type step. Watch Ricardo Lopez.
Your right hand is slow and telegraphed because you leave your right foot behind, Thus it takes a long time to turn your body to throw the punch. Bring your right foot up right away- it shortens your base, gives you a faster right hand and leaves your feet in a position where it is possible to maintain balance and throw more punches.
Finally, that "head movement", leaning the head the the right as you jab is why you are getting hit with right hand counters. Your body should be twisting to the left as you throw your right; your chin, being on your chest should do the same. This will get you out of the way of the right hand counter.