Can we learn anything from observing galaxies edge-on, like the Spindle Galaxy? I was reading about NGC 5866, sometimes referred to as the Spindle Galaxy. It's notable because of its inclination relative to Earth; we see the disk from the side, rather than face-on. The difference is akin to viewing a person's head in profile, instead of looking at their face from the front.

We do lose some information about the morphology of NGC 5866 because of the unusual orientation, as it's difficult to see any spiral arms, bars, or other structures. That said, I'm wondering whether we can still learn more about the galaxy than if we viewed it face-on. For instance, can we gain information about the density distribution of the galaxy at various distances removed from the galactic plane, or perhaps about the structure of its central bulge (if there is one)?
Yes, we can certainly learn things from edge-on galaxies. The easiest things to learn, as you suggest, have to to with the vertical structure of the stars (and gas). For example, our knowledge of how thick galaxy stellar and gas disks are -- and whether this thickness varies with distance from the galaxy center -- comes almost exclusively from studying edge-on galaxies. It is also in many cases easier to study faint stellar and gaseous halos in edge-on galaxies, because in face-on galaxies, these things would be superimposed on -- and largely lost in the glare of -- the face-on disk.

Traditionally, it was also easier to study the radial structure of disks in edge-on systems, because the fact that the line of sight through a disk intersects more stars when the galaxy is edge-on than when it is face-on meant you could make observations at higher signal/noise at large radii. Thus, the detection that many disk are "truncated" (that is, that there is a radius beyond which their brightness falls off more rapidly than it does inside that radius) was first made using photographic data of edge-on spirals.

Some elements of bulge structure have been traditionally easier to study in edge-on galaxies. The realization that some galaxies had "boxy" or "peanut-shaped" bulges -- along with the existence of "cylindrical" off-plane stellar rotation in such bulges and their likely nature as the vertically thickened inner parts of bars -- came about from observations of edge-on galaxies. (Although we now know how to identify such structures in moderately inclined and even face-on galaxies as well.)

(Note that our view of the Milky Way is essentially that of an edge-on galaxy, albeit one seen from the inside.)