Does the Milky Way have any impact on Earth? Does which galaxy we are in (Milky Way) have any impact on Sun/Earth?
What I mean is: suppose our solar system was instantly teleported into a different galaxy (eg Andromeda).  Apart from the stars being different in the night sky, would there be any other significant impact to the processes on Earth?
 A: The stars, or previous generations of stars, have provided most of the chemical elements the Earth, and you and I, are made of. That we are in a galaxy is crucial to our existence.
But I think you mean, is there any impact on the already-existing Solar System? The answer to that is almost certainly yes. The Solar System will be occasionally irradiated by the blasts from nearby supernovae; occasionally a passing star or molecular cloud will get close enough to perturb the Oort cloud and send a cascade of icy bodies into the inner Solar System; the galactic tides may also perturb the Oort cloud. These may have been responsible for mass extinctions in the Earth's history on intervals of $\sim 100$ million years (e.g.Rampino & Stothers 1984; Fernandez & iP 1987; Bailer Jones 2009; Melott & Thomas 2009; Rampino 2015; Fields et al. 2020.
We are also continuously bombarded by cosmic rays, that are accelerated by galactic supernova remnants and confined by the galactic magnetic field.
I don't think putting the Solar System in Andromeda would make much difference to that, since to first order it is quite a similar galaxy to the Milky Way. However, within the Andromeda galaxy (and the Milky Way) there are regions of vastly differing stellar density and vastly different densities of massive stars that might explode as supernovae. Similarly, there are a range of galaxy types, masses and densities - all of which could conceivably have some impact on the rates of the events described above.
A: If our sun was a bowling ball in the hands of the player, the earth would be a grape's seed at one of the pins. The next nearest star (another bowling ball) would be 6000 km away.
So, except for the fact that our starry sky would be different, no difference are expected with average distances between stars like that.
