I.e., is stellar rotation somehow coupled to galactic rotation?

In our own solar system, both rotation and revolution vectors are highly correlated, for sun, planets and satellites. Just about everything has a right handed spin (counter-clockwise) as viewed from the north. This is a clue as to the nature of the accretion processes in the original solar nebula.

If the connection with galactic rotation is there, evidence must be buried deep in the statistics. It is very hard to determine the rotation vector for a distant star, although we note the spin of binary systems seems to be randomly oriented, even in the neighbors of a star cluster. It appears that even among the stars condensing out of one molecular cloud (an open cluster), the spin axes are randomly distributed. My guess is that there is no connection (for example. our ecliptic and the galactic equator are about 60 degrees apart).

Still, if we had accurate data, some interesting relationships might fall out of the statistics. Keep in mind, even in tightly packed globular clusters where stars interact gravitationally enough that they tend to stratify
by mass (the lighter ones are ejected and the more massive collect in the cores) close encounters where their spin axes might be affected are extremely rare.

Also, star formation is an ongoing process, many stars are as old as the Milky Way itself, but new ones are continuously being born.

The galaxy is not just a big rotating wheel of stars. Its pulsing and evolving, gravitationally, dynamically, chemically, morphologically, even shock waves and acoustic phenomena race each other along the disk, even before you add in the result of its interactions with other galaxies.

It's alive.