Velocity of air flow around aerofoil
No, you can't say that the horizontal component of velocity is constant throughout the chord. There are few reasons why not:

1) From the formula you provided (which is correct if your assumption on velocity components is true for an aerofoil) we can see that as beta gets bigger the final velocity tends to infinity, which of course cannot happen.

2)The velocity change on aerofoil is dependant upon its pressure change, it reaches maximum at the point of maximum camber and not at the point of maximum thickness and I think that as per your theory it would than be reached at the point with maximum thickness.

3) Third and probably the most important thing is this: Lets imagine the flat plate aerofoil at some small angle of attack. The plate will produce lift, and the particles above an aerofoil will reach the trailing edge first. Since the airflow above aerofoil will be deflected by the same angle as below the aerofoil, by above stated theory, particles should reach the trailing edge at the same time because of a constant horizontal velocity. Which isn't true.  

Second part of the question is correct, there is no net upward motion of air. In fact, there is a net downward motion of air behind an aerofoil called downwash. Basically how much the force wing exerts on air by deflecting the partincles down, by that amount of force the aircraft will be pushed up. Newton's third law.