Why doesn't the flight path angle increase beyond a certain angle even if the angle of attack is increased?
If you keep increasing your angle of attack, your drag will increase and combined with the increasing along trajectory component (your path angle will increase) of the weight vector will slow you down (assuming your thrust does not exceed your weight + drag). Because you slow down, your lift will decrease. The lift will now be less than the across trajectory component of the weight vector and the flight path will become less steep. 
I don't understand your second point. 
The critical angle of attack is where the maximum lift coefficient occurs. If you increase the angle of attack further, the wing will stall and the lift will decrease. This does not occur at Vx