Let me preface my answer by saying that you're reading a book from 1937 which references the scientific knowledge from the early 1900's. In case you're unfamiliar with astronomical history, I'll say that we've come a looong way since the early 1900's. For example, this book discuses "Nebula as Island Universes". At this point, astronomers had only just discovered that these mysterious "nebulae" were actually entire galaxies that were millions of light years away. Hubble is still referring to them as nebula though, rather than galaxies. That being said, it may actually be kind of difficult to "translate" some of this science from almost 100 years ago into modern theories and terminology. You may even find things which are flat out wrong. Hubble did make a lot of important discoveries, but there were many details he was mistaken on.
Do more recent observations still show a symmetrical nebulae density increase?
Again, nebulae here refers to galaxies, not what we've come to consider as nebulae. What you're asking here, and what Hubble was referring to, before it had an official name, is about the Cosmological Principle.
I think the appropriate answer to this question though, is to point out that I believe you're misinterpreting the text. In the Comparison of Observations with Theory section, Hubble is showing that observations of nebulae (sic, galaxies) indicate they are uniform through space and that his uniformity is easily explained by a static, non-expanding universe.
If both of the extra terms (for recession and for curvature) were absent, the surveys would clearly fit the formula because the situation would be precisely that in a stationary universe.
The "formula" he's talking about here is the one which indicates
nebulae are uniformly distributed through a non-expanding universe in which red-shifts are not primarily velocity-shifts, [and] then the numbers should be proportional to the volumes
However, he also knows that the Universe is expanding (he was the one to discover this after all) so he took the standard formula and threw in universal expansion. What he finds though, is that the equations now predict that
the numbers of nebulae increase faster than the volume of
space through which they are scattered
The important point here is that an expanding universe (with no curvature) predicts this is the case, not that it actually is the case. His previous sections and accompanying plot indicate that observations show the universe is homogeneous. Hubble makes the point (and you reference it) that this is discrepancy can be solved by including the curvature of the universe into the mix as well in which case the predictions now match up with the basic "formula" which didn't include expansion or curvature.
Just as an addendum, I'll expand on this and add that current observations do in fact show that galaxies are isotropic and homogeneous on large scales. One just survey showing this is the 2dF Galaxy Redshift Survey. Shown below is a plot from their results. While full statistical analysis is needed to says something conclusive, you can see that to the naked eye, the galaxy distribution does appear to be homogeneous and isotropic.
Note that this only covers part of the sky because the rest of it is blocked by the disk of our galaxy.
I understand how an expanding Universe explains the symmetrical redshift, but I fail to understand how it explains the symmetrical increase of nebulae density.
I'm interpreting this to mean, you're wondering how the "formula" with expansion included can produce the result that "the numbers of nebulae increase faster than the volume of space through which they are scattered". This I think is more or less a math question. If you're really interested in seeing how it was derived, feel free to look at Hubble's original paper: Hubble, E., Tolman, R. C., Astrophysical Journal, 82, 302, 1935. I think you'll find the summary equations in Section 12 (page 328 in the paper, page 27 in the pdf) with lengthy derivations before that.