Whether or not something is a black hole is dependent on its total mass and its radius. So anything can mathematically become a black hole if you compress it enough. The earth could be a black hole if it's total mass were compressed to something like 'less than the diameter of a grapefruit'. At that point space itself cannot contain the mass, physics breaks down and you get a singularity.
The moon would continue to orbit as it always did, since the moon's centre of mass is still exactly the same distance from the earth's centre of mass as it was before you pressed the 'compress button' on the north pole.
It takes incredible amounts of energy to cause this compression however. And this is why only the biggest stars become black holes. As the outward pressure of fusion diminishes because the hyrogen/helium/lithium/berylium/etc fuel runs out, the sheer gravitational pull of all that mass suddenly takes over and that inward momentum from all directions is enough to cause a singularity.
Not necessarily. If the Schwarzschild radius of the resulting black hole is on the order of a Planck Length you can't really say whether or not such an object is a black hole anymore (depending on your interpretation of the Planck Length).
On my calculator the lower bound is [\frac{r c^2}{2G}] which is ~1.09e-9 kg (give or take a factor of two depending on whether you want to consider the diameter or radius). Which is small but not as small as you might think. I also believe there are some upper limits on black holes too (that come from the upper limits of stars).
The sun is not pulling in the earth. Earth orbits a common barycenter with the sun. That barycenter happens to be inside the sun, but the sun is also orbiting around it.
So a small object orbiting a bigger object is an approximation. Orbiting their common center of mass is a better approximation. This is more easily visible in the pluto-charon system.[0]
The same of course applies to black holes. Them being heavier than most other objects in their vincinity does not render them immobile any less than the sun being the most massive object in the solar system renders it immobile.
[0] https://en.wikipedia.org/wiki/Charon_(moon)#/media/File:Plut...