If it doubled in mass every hour once it started falling, it wouldn't take that long to grow.
It doesn't, that's the point.
As it got bigger, its growth rate would increase, because it would be exerting more force.
Of course, it's a function directly linear with the square of mass intake, since the schwarzschild radius increases directly linear to the mass of the black hole. Doubling the mass means doubling the radius which means twice the surface area and twice the mass intake.
Plus, pressure inside the earth would force mass against it. Not as strongly as within the sun, where the pressure causes fusion, but probably strong enough to grow the singularity.
Completely irrelevant here, since a proton-mass black hole has a radius much smaller than a single proton. Unless you have pressure high enough to create neutronium, it doesn't increase the mass density by any remotely relevant factor. Which is the reason why a subatomic black hole grows extremely slowly. It spends all its time, zipping around between nuclei and electrons without ever hitting anything, like an asteroid in a solar system. Until it reaches at least atom-size, the growth rate is seriously frickin slow. The "density" of a black hole is on the order of 10^10 of times higher than the density of an atomic nucleus, let alone an atom. And at first, the thing wouldn't even be able to swallow a nucleus whole. It hits a neutron, and the rest of the nucleus goes pop.
I suspect Hawking is wrong about the mechanism for BHE. I think the gravitational field of a BH perturbs space, which gives rise to spontaneous particle generation. Gravity does work, so it must transmit energy. So, a black hole bleeds energy by the gravitational work it does on other objects, or more likely on the fabric of space itself. All objects do, but it is so small we haven't learned to measure it, because gravity is so weak.
Well how exactly black holes evaporate doesn't matter, as long as they do. If they do, smaller and smaller black holes quickly reach a point where they evaporate faster than they can accumulate mass. If Hawking is correct, that mass is several tons, not a couple of AUs (atomic, not astronomical units). In this case, the last few tons of a black hole would also get converted to energy and particle radiation in a spilt second, like one hell of a nuke. (or antimatter explosion)
If black holes evaporate because of gravity, the size limit for sustainability would need to be calculated, but it would certainly be much, much higher than a subatomic size black hole.
Read Prey by Michael Crichton. Nanobots released into the environment haave access to raw materials.
I'm afraid to say that Michael Crichtons approach to science is about the same as Dan Browns approach to cartography.
The problem with nanobots is that changing molecules around takes energy, and nanobots being really, really small can't very well carry around gas tanks. It's not a problem for von Neumann cybernetics in general, just for it working at a nano scale.
So, what is the S-radius of the moon? And what is the S-radius of the maximum-energy collision from the LHC?
The radius of a black hole with the mass of our moon is around 10^-5 meters. Can't say exactly for the LHC, i don't know what the highest possible resulting mass from a collsion experiment would be. Let's call it a few thousands or tens of thousands of neutron masses. So we'll get a black hole that's ~10^6 times smaller than a single proton.
For that matter, what is the S-radius of a quark or an electron? Could this explain how they are self-sustaining?
I'm not quite sure what you mean. Since neither quarks nor electrons are black holes, i don't know what you mean by them being self-sustaining.
If so, it would put an upper bound on their physical Size.
We have upper bounds for the physical sizes for those. (around 10^-20 meters iirc, +- a few orders of magnitude, quarks are bigger) But those are derived from various experiments.