Your second paragraph is mostly right. But let's be a little more precise because it's easy to be misled. B
You are right that at about 50 to 100 parsecs you see pretty strongly the statistical homogeneity and isotropy, and can measure the expansion of the universe from the redshift due to the velocities of galaxies at those distances. In fact, you can do better, and even at a few Mpc and for sure 10 you can see the expansion, with more distant galaxies receding from us faster than those closer in. See for instance a graph of the Hubble velocity/distance graph for relatively close in galaxies, as shown above. But also notice the non statistical spread in the blue ellipsoidal area: those are different galaxies, 'HELD' together by their self gravity in the Virgo Cluster. Averaging those things out it gets more consistent at larger distances. That is from the wiki articles on the Hubble expansion at https://en.m.wikipedia.org/wiki/Hubble's_law#/media/File%3AHubble_constant.JPG
Hubble has to do it with much closer in data, un to 2 Mpc, that is what he could confidently measure in 1929. See his plot right below. The expansion was thus suspected and accepted to some extent, but not really fully unit more measurements were done. The coup de grace was the discovery of the CMB radiation indicating that there had been a Big Bang.
The local effects dominate because the gravitational fields from nearby galaxies are just much stronger than that from much further away, even if there's a lot of them. You see that in the data for the Virgo galaxies in the first figure above. As for the earth, the gravitational pull of the Sun is more than that from other stars in the Milky Way, or nearby galaxies like Andromeda. We do have some galactic neighbors, and are part of a galactic cluster. All of those have some influence and in fact our whole galactic cluster has a peculiar velocity with respect to the average cosmological expansion. We have to subtract the effect of the earth's peculiar velocity with respect to that average motion to make sure our CMB measurements are then in a coordinate frame that is comoving with the universe, and it is from which the universe looks isotropic and homogeneous, on the average. The CMB has been probably the best way to get to that peculiar velocity which then makes us, after subtracting it, see the CMB as homogeneous and isotropic as possible. That peculiar velocity is about 360 km/sec in the direction of the constellation Leo. See the parts that have to be added together to obtain out total local velocity with respect to the average motion of the universe's expansion at http://image.gsfc.nasa.gov/poetry/ask/a10552.html