Do the axes of rotation of most stars in the Milky Way align reasonably closely with the axis of galactic rotation? The axis of rotation of the Solar System makes a large angle of about 60 degrees relative to the axis of rotation of the Milky Way.  That seems unusual - for example, most of the bodies within the  Solar Sytem are better behaved than that.  
Do most stars or planetary systems in the Milky Way rotate in close accord with the galactic rotation?  Or is there a large scatter, so that, in fact, our Sun is not atypical?
There is very likely to be a random scatter.
Unlike planets orbiting the Sun in the Solar System, most of the stars in the Galaxy did not form at the same time as the Galaxy itself. There is therefore no strong reason to suspect that the angular momentum vectors would be aligned for similar reasons. On the other hand, the Galactic gravitational potential does depart from spherical symmetry in its inner regions, because the visible matter, which becomes dominant in the inner regions, is concentrated into a disc - so presumably this, or perhaps the tidal forces exerted by this on molecular clouds, could imprint some angular momentum preference.
The evidence is sketchy but suggests random orientations, at least in the solar neighbourhood. I refer you to Detection of exo-planets , where I discuss this in the context of detecting transiting exoplanets.
In a series of papers, myself and colleagues have investigated the distribution of spin-axes in clusters of stars. The idea here, which is not far-fetched, is that large clouds from which clusters form will have some angular momentum. The question is how much of that angular momentum is inherited by the stars it forms, or to what degree can turbulence in the collapsing gas essentially randomise the spin vectors of collapsing fragments. Our technique was to combine rotation periods (latterly from Kepler observations) with careful measurements of projected equatorial velocities ($v \sin i$, where $i$ is the spin inclination to the line of sight) to get projected radii ($R \sin i$) and then to model the distribution of $R \sin i$ with various assumptions about the spin-axis distribution. In all three of the clusters we have studied (Pleiades, Alpha Per, Praesepe), the distribution was consistent with a random distribution, with quite strong limits on the amount of alignment that was possible (Jackson &amp; Jeffries 2011; Jackson, Deliyannis &amp; Jeffries 2018; Jackson et al. 2019). The technique has been replicated in a fourth cluster, NGC 2516, by Healy &amp; McCullough (2020), with the same conclusion.
Other authors have claimed alignments in some cases. Notably, using Kepler asteroseismology of red giants in two clusters in the Kepler main field, Corsaro et al. (2017) claimed a quite tight alignment of spin axes, pointing almost towards us in each case. Since the Kepler field is not far from the Galactic plane and these were distant clusters, then the spin axes would be almost in the Galactic plane (a bit like Uranus and the Sun). However, the likelihood of finding such a result if individual clusters had random average angular momentum vectors raised question marks - the probability of seeing that vector pointing towards you is very low. Work by Kamiaka et al. (2018) shows that the asteroseismological estimates may be systematically biased towards low inclinations.
A further piece of evidence for some alignment was in the orientations of bi-polar planetary nebulae towards the Galactic bulge. Rees &amp; Zijlstra (2013) found a non-random distribution that suggested that the orbital angular momenta of binary systems, responsible for the bipolar shape of the nebulae, were oriented aligned with the Galactic plane (again, like Uranus around the Sun). The result is highly statistically significant but as far as I know has not been followed up despite its obvious implications for estimations of transit yields from exoplanetary surveys.
I think the biggest argument that there is no significant effect for average stars in the field of the Galaxy, is that the exoplanet people working on the TESS survey (which covers the whole sky), would have found a drastic spatial dependence on their yield of transiting planets as a function of Galactic latitude. The majority of transiting planets (or at least hot Jupiters) have orbital axes coincident with the spin axis of the star (like planets in the Solar System). If these orbital axes were aligned with Galactic north (or any other direction) it would mean you would see far fewer transiting planets when looking towards those directions. I have heard no reports of such a spatial dependence.