Why can a sharp surface ionize the air molecules?

Answer 1

The dependence of electric field strength around a point charge has the form E = charge x a_spatial_dielectric_constant / d (distance away from the surface)^2 .

So, when you have a surface of very small radius of curvature, the field close to the surface is enormous, since d is very small. That enormous field can spray electrons onto adjacent air molecules (if negative sign), or pull them off (if positive sign).

Sharp (or more specifically, pointed) objects excel at this, that's why lightning rods are pointed on top (they initiate an ionization which either starts the bolt or receives it gladly. Similarly, fine-wire copper brushes are used to ionize room air negatively (remember: negative ions have positive effects!) for air-cleaning and immunity-boosting purposes.