Why are stars so far apart? So, this is mere musing, but it seems that stars are quite extremely far apart. I tried to determine the mean distance between nearest neighbors for stars (in just our galaxy) but I'm not sure what it is, though a first approximations I would guess that the mean is around 5 ly with some 0.5 standard deviation or so!

However, assuming I didn't screw up, the distance between two sun-massed objects required to generate 1,000N of force between them is an incredible 54.3 million ly. [Edit] If you change that to 1 billion Newtons, the distance is one galactic radius -- 54000~ ly!

I'm probably mis-estimating how much even 1kN of force would effect solar trajectories over time, but I still can't help but feel that our solar neighbors are very far away given the spheres of interaction that I would expect.

Any insights?
Most stars are of a solar-mass or below. The average number of companions that each stars has (in the sense of being part of binary or higher multiple systems) systems ranges from 0.75 for stars of a solar mass to approximately 0.35 (not a well-established number) for the more numerous M-dwarfs. Let's take a compromise value, say 0.5.
The separation distribution of these multiples peaks at around 50 AU for solar-type stars, reducing to about 5 AU for low-mass M dwarfs. Again, lets take a compromise value of 20 AU. See Duchene &amp; Kraus (2013) for all the details.

So if we take 1000 stars, then 333 of them (roughly speaking) are companions to another 333 stars, while 333 are isolated single stars. (NB This does not mean the frequency of multiple systems is 50%, because some of the companions will be in higher order multiple systems)

Thus, taking your calculation of the separation between stellar systems of 5 light years ($= 3.2\times10^{5}$ AU), then the mean separation is:
$$\bar{D} = 0.667\times 20 + 0.333 \times 3.2\times10^{5} \simeq 10^{5} AU, $$
but the median separation is 20 AU!

This is of course sophistry, because I'm sure your question is really, why are stellar systems so far apart?

Stars (and stellar systems) are born in much denser environments. The number density of stars in the Orion Nebula Cluster (ONC - the nearest very large stellar nursery) is about 1 per cubic light year. The equivalent number for the solar neighbourhood is 0.004 stars per cubic light year. Thus the average interstellar separation in the ONC is 1 light year, but in the solar neighbourhood it is about 6.3 light years.

The reason for this separation at birth is the Jeans length - the critical radius at which a clouds self-gravity will overcome its thermal energy and cause it to collapse. It can be expressed as 
$$\lambda_J = c_s \left( \frac{\pi}{G\rho } \right)^{1/2},$$
where $c_s$ is the sound speed in a molecular cloud and $\rho$ its density. For star forming giant molecular clouds $c_s = 0.2$ km/s and $\rho=10^{-23}$ kg/m$^3$. So clouds of scale hundreds of light years could collapse. As they do, the density increases and the Jeans length becomes smaller and allows the cloud to fragment. Exactly how far the fragmentation goes and the distribution of stellar masses it produces is an area of intense research, but we know observationally that it can produce things like the ONC or sometimes even more massive and dense clusters.

From there we know that a new born cluster of stars tends not to survive very long. For various reasons - outflows, winds and ionising radiation from newborn stars are able to heat and expel the remaining gas; star formation appears to have an average efficiency of a few to perhaps 20-30%. Expelling the gas, plus the tidal field of the galaxy breaks up the cluster and disperses it into the field, which gives us the lower field star (or system) density that we see around us.

Once the stars are part of the field they essentially don't interact with each other; they are too far apart to feel the influence of individual objects and move subject to the overall gravitational potential of the Galaxy.
 so your consideration of the force between stars is not really relevant.