Are event horizons "geodesic"?

An event horizons is always generated by null geodesics. Because it is defined as the boundary of the past of future null infinity $\mathscr{I}^+$, it is a null hypersurface, which is always generated by null geodesics (i.e. define a null geodesic congruence). The only subtlety you have to worry about are caustics, which are where null generators enter the event horizon, and make the horizon look less smooth at certain points. But these do not spoil the fact that every point on the event horizon lies on a null geodesic that remains on the event horizon indefinitely to the future.

Some discussions of this can be found in Wald on/around pg. 311.

Yes, under certain conditions. A theorem due to Hawking states that in a stationary, analytic, asymptotically flat vacuum black hole spacetime, the event horizon is a Killing horizon, which in particular means it is a null hypersurface, which in particular means it is a null geodesic congruence.

EDIT: asperanz's answer below is better, in my opinion.