How was the galactic plane established? As pointed out in this answer the axes of galactic coordinates are defined such that galactic longitude and latitude (0, 0) point toward the galactic center. But then the plane defined by galactic latitude equal to zero would be parallel to the galactic plane only if the Sun were in the galactic plane as well. 

Is that the case? 

More generally, by what procedure was the galactic plane actually defined, considering a substantial part of it is obscured by dust (at least in the visible) and most of the mass is dark matter and therefore not observed at all (directly at least)?

Question: How was the galactic plane established? Primarily what kinds of observations were used, and how were they analyzed to help decide how this plane should be chosen?
The zero of Galactic latitude (i.e. the Galactic plane) was defined by a working group of the International Astronomical Union. The observations used to do this were 21 cm radio observations of atomic hydrogen, which are unaffected by dust absorption near the plane. The details of the observations are found in Gum, Kerr &amp; Westerhout (1960). This paper uses least squares fits to the 21cm data assuming that the greatest concentration of neutral hydrogen is found in the Galactic plane itself. Whether this is the dynamical plane was considered by Gum &amp; Pawsey (1960), as part of the same IAU study. They concluded that it was (bar the location of the Sun with respect to the plane), citing the flatness and relative narrowness of the neutral hydrogen and the coincidence of the source Sgr A*, which even then was thought to be at the Galactic centre, with zero latitude.

Dark matter was not well explored or widely considered in the dynamics of individual galaxies in the late 1950s, however it is not unexpected that neutral hydrogen should lie in the Galactic plane, as defined by the symmetry of the Galactic potential. A rotating mass of gas, capable of dissipative interactions, should settle to a plane perpendicular to its angular momentum vector.

The 21cm definition of Galactic coordinates cannot itself define the position of the Galactic plane in 3d space unless the Sun is exactly in the plane. Detailed observations of star counts and the dynamics of stars around the Sun and perpendicular to the plane have revealed this isn't quite the case. The details are given in How far is the Earth/Sun above/below the galactic plane, and is it heading toward/away from it? ; the Sun is currently around 17 pc above the Galactic plane and heading upwards. This is close enough that it makes very little difference to the apparent flatness and narrowness of the 21cm data.