This is relevant to the discussion at hand because for these very large black holes that density is not very high and conceivably a gas cloud of sufficient mass could contract to that density and have an event horizon form without collapsing into Stars.
No, it doesn't. There are indeed upper limits on the mass of objects that are formed from stars that run out of nuclear fuel, before the objects collapse to black holes (the Chandrasekhar limit for white dwarfs and the Tolman-Oppenheimer-Volkoff limit for neutron stars), but those limits are not based on density.
It is true that, especially in the early universe, gas clouds forming very massive black holes without going through the intermediate stage of forming stars is considered possible; but that is not based on the kind of simplistic calculation that you describe. It's based on numerical simulations of the Einstein Field Equation with relevant initial conditions.
Another factor you are not considering is that the universe is expanding, and the early universe was expanding much more rapidly than our current universe is. So gas clouds contracting to form very massive black holes had to work against the expansion to do so. That further complicates the calculations.
Is that correct? My understanding is that the expansion of the Universe occurs away from large concentrations of mass; expansion doesn't cause the stars in a galaxy to move apart.
...is not what a black hole is. It is not even "something kinda sorta like a sphere with some Schwarzschild radius". It is nothing like any ordinary object you're used to. It doesn't have a well-defined "radius" any more than it has a well-defined volume; the Schwarzschild "radius" is actually sqrt(A / 4 pi), where A is the surface area of the hole's horizon (which in turn is calculated from the mass of the hole).