Can a steady flow have stagnation points?
The idea of a stagnation point is an idealization. This point is infinitesimally small, and air particles flowing along a streamline which leads into it will slow down on their way. The closer they come to the stagnation point, the slower they flow, and in the end they never arrive at the stagnation point.

In reality, air molecules have a finite size, so they flow either above or below the stagnation point streamline. Even if one particle manages to arrive at the stagnation point and stays in place (which is not possible in theory), a small angle of attack change will wash it away in the next moment.

One streamline goes away from the rear stagnation point which is located in the trailing edge. This point is doubly idealized, because it requires inviscid flow to have a rear stagnation point. Again, molecules flowing along the wall of the airfoil will be slowed down by the stagnation point pressure the closer they are to the trailing edge. Since they arrive either above or below the rear stagnation point, they will accelerate once they have passed the point of highest pressure and move on above or below the streamline emanating from the rear stagnation point.

The concept of a stagnation point is really helpful for understanding flow phenomena, however. There is really a line (in 3D it is a plane) which separates air that will flow over the wing from that which will flow below it. This line changes with angle of attack, and the simple stall warning vane is based on this principle. Once the line ends below the vane, air will press it upwards, closing an electrical contact which activates a buzzer in the cockpit.


Stall warning vane (small metal thingy sticking out from the wing)