Number of galaxies per redshift

The term "a distribution of number of galaxies per redshift interval" does make sense, but usally we restrict ourselves to a certain type of galaxies, e.g. Lyman $alpha$ emitters, Lyman break galaxies, sub-mm galaxies, etc. The reason is that galaxies are detected in a number of different ways, and no single selection technique captures all types of galaxies.

Moreover, since galaxies have no well-defined lower size/mass/luminosity threshold, we have to define a threshold above which we count them. For this reason, rather than talking about "the number density of galaxies per redshift", we usually use the very useful observable luminosity function (LF) which measures the number of (a certain type of) galaxies per luminosity bin, and per comoving volume, at a given redshift.

LFs have been probed out to very high redshifts, at the times when galaxies were just beginning to form at $sim$half a gigayear after Big Bang (e.g. Schenker et al. 2013).

You end your question by asking about the number of galaxies at $zle0.01$, which would be considered very low redshift in the context of galaxies. You can find the answer by considering all kinds of selection criteria, checking for overlaps between surveys (most local galaxies will be detected by multiple techniques), and integrating over volume the LFs down to your chosen threshold (e.g. Small Magellanic Cloud-sized). Doable, but not trivial. An alternative is to integrate numerically calculated dark matter halo mass functions, using e.g. this online tool. I get a number density of $nsim0.4$ halos per $h^{-3},mathrm{Mpc}^{3}$, or $1.2,mathrm{Mpc}^{-1}$. The comoving volume out to $z=0.01$ is $V=3.3times10^5$ Mpc, so the total number of (super-SMC-sized) galaxies is $N = n V simeq 3.8times10^5$.