These are called the black-hole normal mode decay rates, the decay is analogous to a membrane with a certain viscosity type friction which is dissipating it's energy, and the rate of exponential smoothing out of surface fluctuations is the rate at which the black hole equilibrates. The identity of the two processes, the membrane-paradigm surface smoothing from dissipative dynamics and the string-theoretic equilibration of the dual field theory, was a major motivator for the holographic principle in the 1990s. The normal mode rates are calculated in several places, I don't remember the rates offhand.
Regarding "making cusps", a steady insertion of matter flowing in a line-stream into a black hole should make something like this, but I never saw the precise form of the metric, or the deformation of the horizon, but it should make a cuspy structure of some kind. There are other singular constructions which are easier, involving time-reversal of black hole merger, but I don't remember how they work. Another cusp is the path of transition between a 5d linear black hole and a line of 5d sphere black holes, the Gregory Laflamme instability. The 5d black hole has to break up, but the way it does so is by producing many black holes linked by thin horizon tubes, which are singular cusps at the point where they connect with the large horizons. The scaling limit of this is computable, but it has never been done in the literature.