Assuming an exact black hole solution to the Einstein Field Equations (EFEs), what's happening is that any observer sufficiently close to the singularity will inevitably collide with the singularity. One can then define the horizon the outer boundary at which the future light cone attached at every point ends in the singularity. In order to avoid the singularity in the future one would need a causal cone broader than that of the light cone. The boundary is only a surface in the abstract; it's not material, you can't touch it, nothing bounces off it, and the no-drama conjecture says that if you were to float through the horizon you won't immediately notice you've done so.
Outside the horizon as defined above, some part of the future light cone will have the singularity in it, however as one choose an observer at a point further and further away from the black hole, the singularity occupies a smaller and smaller part of the future light cone for that observer at that point, and the observer would have to aim ever more carefully to collide with it. (The observer simply has to reach a point at the horizon).
> a black hole has nothing underneath the event horizon
That's true of vacuum solutions of the EFEs which contain black holes, but it's not true in general. The Vaiyda metric is an exact solution which equips a spherically symmetrical central mass with spherically symmetric non-interacting dust of incoming and/or outgoing radiation: a Vaiyda black hole can have some incoming radiation within the horizon.
Astrophysical black holes are more complicated since the matter outside them is generally far from spherically symmetrical. However, within the horizon will be some of the cosmic microwave background's (CMB's) photon gas, and whatever else reaches the horizon.
But, you're right that something must be deep inside an astrophysical black hole, and the something is perhaps not a classical singularity, but rather some configuration that preserves the quantum mechanical properties of the field-content that reached the horizon.
> might evolve over time
If there's infalling matter, it must evolve over time, as the mass will increase, and likely so will the angular momentum, and charges are likely to go back-and-forth as one cannot guarantee that only chargless matter (e.g. photons) will fall in, nor that if we let charged matter fall in, it will always be charge-neutralized (e.g., no ions allowed!). That's true of a classical singularity, and it will be even more true for any ultradense non-singularity structure that preserves QFT information.
> faster-than-light cascade of space-time
What's that?
> Gravity is affected by curvature
In General Relativity, gravity is curvature.
> whatever might be in the centre cannot be the cause of the event horizon
Mass-energy generates the metric. In the case of the Schwarzschild black hole solution to the EFEs, the central mass (which is all the mass and energy in the vacuum Schwarzschild spacetime) is static and concentrated at one point. However, a non-rotating star, or a slowly-rotating object like the moon, generates a very close approximation of the Schwarzschild solution. Astrophysical black holes generate an approximation of the vacuum Schwarzschild black hole spacetime, but the Kerr metric is a better approximation for rotating black holes, and there is plenty of matter in the universe outside black holes. The true metric of the entire universe is much much more complicated than our (useful!) approximations, but nevertheless it is determined by the scatter of mass-energy throughout the spacetime. The true metric almost certianly describes regions which behave almost indistinguishably from the interiors of Schwarzschild (and Kerr, etc.) black holes.
Another way of looking at this: as one throws matter into a black hole, the mass of the black hole increases, and so therefore does the diameter of the black hole. If you allow matter to cross into the black hole and collide with the singularity (or whatever replaces it), how can you maintain the italicized part of the quote above?
> the event horizon is stable in itself
If you allow mass to increase by letting dust and gas and cosmic microwaves reach the horizon, how do you keep the horizon stable?
Moreover, if you merge two black holes, how do you keep both black holes' horizons stable?
> It's a self-sustaining
Given Hawking radiation then a black hole that's a solution to the Einstien Field Equations and equipped with one or more quantum fields filling the whole spacetime where the energy-density of the quantum fields is below a critical threshold will shrink. In the limit of quantum vacuum outside, this shrinking becomes complete evaporation: the horizon simply is not found in the future, because the mass in the whole spacetime has reconfigured from concentration at a central point to diffusion away from the central point. Since mass-energy generates the metric, the metric ceases to be Schwarzschild (or whatever black hole metric had existed prior to final evaporation).
In our universe, there is lots of mass-energy outside black holes, and even though the metric expansion of space will make the CMB energy-density very small, it will still be higher than the critical threshold for stellar mass and larger black holes for a lonnnnnng time. But this isn't the black holes self-sustaining: they would evaporate, except for the CMB dust falling onto them, so it's the CMB (and its neutrino equivalent) that's doing the sustaining.
