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Can the event horizon of a black hole vibrate? If so, are there mechanisms that dampen the vibration?

Consider a spherical, non-rotating, non-charged black hole located far away from other sources of gravity.

Suppose a spherical star having mass M(<10% mass of black hole) approaches on a collision course toward the exact center of the black hole.

In general, boundary implies some sort of equilibrium. So I'd expect the Event Horizon to start off smooth. At equilibrium I'd expect perturbations like this to induce roughly simple harmonic oscillations. Does that happen?

If vibrations are induced, are they dampened and if so, by what mechanism? Hawking Radiation? Gravitational Waves?

Are there expected resonance modes?

Are there ways that properties of the original star be "imprinted" on the event horizon? If there's no dampening, then I'd expect there to be distortions at the direction of approach.

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Yes horizons can vibrate. The resulting damped oscillations are called "quasi-normal modes." There is a large literature on them: see
arXiv:gr-qc/9909058 for a review.

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