Yes. The radius of a black hole increases linearly with mass. Supermassive black holes can actually have really small average densities.
> [SMBH]s can actually have really small average densities.
No kidding.
Forgive me if you know all this next three paragraphs.
Christodoulou & Rovelli (2014) (Phys. Rev. D 91, 064046 doi:0.1103/PhysRevD.91.064046) [C&R 2014] show the volume of a spherically symmetrical non-eternal BH grows significantly with time (eqn 1 of ). "The bulk of the volume turns out to be due to a region in the vicinity of a constant value of the radial coordinate. That is: inside the hole there is a long spacelike 3d cylinder with slowly varying radius, which grows longer with time ... For instance, the black hole [Sag Astar] has radius ~ 10^6 km and age ~ 10^9 years. Inside it there is space for ~ 10^34 km^3, enough to fit a million Solar Systems!" (emphasis theirs)
[C&R 2014] provoked several other papers including Begntsson & Jakobsson (2015) (Mod. Phys. Lett. A, vol 30 no. 21, doi:10.1142/S0217732315501035) extends this to black holes with significant angular momentum, whose interior volumes are much smaller, with [C&R 2014]'s non-rotating volume being an upper limit. Their results are amplified and some implications thereof considered by Ong (2015) (J. Cosmol. Astropart. Phys 04(2015)003 doi:10.1088/1475-7516/2015/04/003), delightfully titled "Never Judge a Black Hole by Its Area", which examines several other types of BH too. In particular Ong clarifies that the [C&R 2014] results are likely robust for astrophysical near-extremals like Sag. Astar.
Of course, these are studies of theoretical BHs (and mainly considered statically), but I'm pretty confident that typical stress-energy around mature astrophysical black holes won't perturb dramatically away from that.
If you like, I can quickly check with an expert on astrophysical black holes for the case of a close binary where the mass flow from the donor star is very large, or for very young BHs that still have most of their progenitor star outside the horizon, but with agreement that such a system is far from what we would have when swapping our sun with a solar-mass BH.
> Yes.
It really depends on whose definition of compact object you take. I think the majority view -- bearing in mind this is an astrophysics classification rather than a relativity one and I am in the latter camp -- is that compact objects are characterized in one or more of three ways: (a) the presence of strong gravity in the vicinity, (b) hierarchical or direct formation from gravitationally collapsing matter, (c) the presence of matter densities much higher than that of planets.
A quick survey of observational astrophysical groups (Harvard, MIT-Kavli, Cambridge-Kavli, Toronto, KTH), fails to find one that both [i] unambiguously splits supermassive black holes out from compact objects (Harvard comes close), and [ii] restricts their work to small black holes that would meet (c) in the way you want (in fact, all of the groups I quickly looked at explicitly work on SMBHs as well as NSs, white dwarfs, and stellar BHs, but perhaps my selection is unfairly biased).
Astrophysicists and relativists can be surprisingly different beasts, but I think few of the former would insist on inserting "mean" into (c) and then exclude very massive BHs as not compact on that basis, especially given [C&R 2015] and subsequent results for spherically symmetrical non-rotating stellar-mass BHs.
It also seems like it would be a pretty strange exclusion since SMBHs -- however you define their mean density -- certainly exhibit relevant-to-astrophysicists general-relativistic phenomena near the horizon, and would still do so even if (for a sufficiently massive BH) the curvature just outside the horizon falls below the curvature we experience here on the surface of Earth.
Finally, I object if your definition of average density depends on any definition of internal volume, as that would depend on a choice of spatial section, which in turn depends on a choice of coordinates, and thus cannot be other than an observer-dependent quantity. In general, an observer can be found that makes a complete mess of one's expectation for such a quantity.
(ETA: Ong makes exactly this point at the even more delightful discussion of his paper op. cit. at https://www.kth.se/profile/ycong/page/the-interior-volumes-o... )
