Horizontal stabilizers stall, so I was wondering, is it possible for an aileron to stall? If yes, at what angle of attack?
The aileron itself doesn't stall, being part of the main wing. But since it alters the camber of the main wing it changes the local angle of attack in effect. On older airplanes, a down aileron at low speed could trigger a stall separation of the wing in the area of the aileron and if you applied aileron to correct a wing drop at the stall, it would just speed things along and over you go. This was one of the reasons for slots in front of the ailerons on airplanes like the Globe Swift.
Not sure how it is taught today (I learned to fly in 1975) but long ago pilots were always taught to avoid aileron input close to the stall and keep wings level with rudder, although on modern airplanes with washout and other features to delay stall near the tip you can generally get away with it.