For example, it is possible to have a lighthouse whose beacon is spinning so quickly that the spot of light in the distance (at some distance "r") is moving faster than c, even though the rotation of the lamp is not faster than c, nor is the actual light beam.

Each individual photon is travelling at c and, ignoring nearby large masses, is travelling in a straight line. The fact that the next photon emitted by the beacon is pointed in a new direction that seems to be a long way from the prior emission is immaterial. The photons are independent.

Where it gets interesting is if you have a beam of electrons travelling in a circular orbit at very close to the speed of light. Because they are changing direction as they go around the storage ring they are accelerating and they emit photons as a result. In the rest frame of the electron the emission is isotropic but in the rest frame of the lab there is a Lorentz transformation which focuses all the light into a narrow cone in the forward direction so that the light is always at c. This is known as synchrotron radiation and is a valuable tool for studying matter (or an annoying waste product if you are a high energy physicist).

c.f. en.wikipedia.org - Synchrotron radiation

Rgds,
HH/Sean