In computing the electric field at P, the video considers the middle charge and says it points upward. The rightmost charge points upward and to the left. The leftmost charge the electric field points downward and to the left so if you draw that out you'll be able to find force on P.
During the time interval you mentioned, he temporarily omitted the middle charge because he is calculating the contribution from the other two charge, which turns out to form a dipole. You should see toward the end of the video he stated that the net field would be the total field from the middle charge and the dipole.
No, it applies for any value of ##r##. The condition of ##r>>a## was imposed in order for the expression of the field due to the dipole to be able to be written in terms of ##r##, instead of the distance from the right (or left) most charge to the observation point.