Why do positive and negative electric charge so exactly cancel for large objects? Why is there no appreciable electrical force between, say, planets?  I learned long ago that the electrical force between a pair of electrons is about 42 orders of magnitude greater than the gravitational force.  Between a pair of protons the difference would be ~36 orders of magnitude.  Yet the gravitational force between, say, the Earth and the Moon dominates over the electrical force, so much so that I've never heard of anyone even considering the electrical force.  Apparently the number of positively charged particles and the number of negatively charged ones making up each body are numerically equal to within perhaps 1 part in 10^45 or better.  This seems like a fantastic coincidence. The total number of protons/electrons making up the moon is only maybe 10^50.  I can see that a planet with a net charge would tend to expel particles of the same charge.  But then where do those expelled charges go?  Are there lots of protons (or electrons, or other charged particles) flying around free? If so, are they mainly positive or negative? If not, then presumably the total number of positive and negative charges in the universe are equal to inconceivable precision.  Why would this be the case?
 A: Consider two planets each of mass $M$ at a distance $d$ from eachother, each with charge $Q$. The acceleration the planets exert on one another due to the electromagnetic force would be
$$a = \frac F m = \frac{Q^2}{Md^2}$$
Let's suppose this acceleration is astronomically appreciable, say $1\, \mathrm{m}\, \mathrm s^{-2}$. This means $Q\approx d \sqrt M$. 
Now let's consider how that kind of acculumated charge would interact with something on the surface of one of the planets. If the net charge on the planet is distributed evenly, then the acceleration of an electron on the surface would be
$$a = \frac F m \approx 1.76 \times 10^{11} \mathrm{A}\, \mathrm s \,\mathrm {kg}^{-1}\times\frac{d \sqrt M}{r}$$
where $r$ is the radius of the planet. Let's say the radius of that planet is $6,371 \,\mathrm{km}$, the interplanetary distance was $10 AU \approx 1.5 \times 10^{12} \, \mathrm m$ and the mass is $6\times 10^{24} \, \mathrm{kg}$.
$$a \approx 1 \times 10^{29} \, \mathrm{m}\, \mathrm{s}^{-2}$$
Oh dear. I don't think any loose charges on the planet will be around for long.
NB: The planet's mass and radius were chosen to match Earth, whose gravitational acceleration on the surface is about $10 \, \mathrm m \, \mathrm s^{-2}$. So the electromagnetic force on an electron in this case was $28$ orders of magnitude greater than the gravitational force.
A: The standard model of particle physics developed slowly from the macroscopic observations of classical physics  to the microscopic where quantum physics has to be used. The macroscopic observations up to the twentieth century showed electrical neutrality for the bulk of matter, unless an effort (energy expended) was made to create charge . So "bulk matter is neutral" unless rubbed was a postulate for classical physics. This as carried on to the first quantum mechanical models, of atoms made up of electrons and protons and neutrons, to the standard model of quarks and leptons we have presently as THE mathematical model of nature. This model works and is based on postulates as: charge is conserved, baryon number is conserved, lepton number is conserved, etc.
So charge conservation, and baryon number conservation  with the basics of the standard model ensure that bulk matter will be neutral, and to separate charges energy has to be expended.
As classical physics emerges from the underlying level of quantum mechanics, the absolute postulates of the standard model, since it is so very well validated, ensure that bulk matter will be neutral. 
There exist hypothesis for charged black holes, for example:

Since the electromagnetic repulsion in compressing an electrically charged mass is dramatically greater than the gravitational attraction (by about 40 orders of magnitude), it is not expected that black holes with a significant electric charge will be formed in nature.

So another observational reason for lack of charged moons or stars is that the formation of celestial bodies   fits very well the classical gravitational model of gravitational attraction in the primordial plasma which  in the end turns  into galaxies and stars and planets around the stars. As like charges repulse there would be no coagulation due to gravity being so weak: the atoms would be repulsed by the same charges long before the very much weaker gravity could pull them in.  The very existence of planets argues that they have to be neutral.
So, starting with overall neutrality at the quantum level, neutrality is the rule macroscopically  unless energy is expended to separate charges.
A: 
Apparently the number of positively charged particles and the number
  of negatively charged ones making up each body are numerically equal
  to within perhaps 1 part in 10^45 or better. This seems like a
  fantastic coincidence.

This isn't a coincidence.  It's a consequence of mobile charges.  It is the same reason that astronomical objects above a certain size are round. 
A large charge imbalance or a large mountain both represent a source of potential energy that can be released by moving things around.  As you mention, electromagnetism is relatively stronger than gravity.  That means there can be less of a charge imbalance before fields become so strong that charges are ejected.  The stronger the force, the more energy available by a small imbalance.

Are there lots of protons (or electrons, or other charged particles)
  flying around free? If so, are they mainly positive or negative?

In this region of the solar system?  Yes.  The solar wind consists of lots of charged particles.  The net charge of the wind is neutral, so both species are present.
Imagine that an object in this region had a charge imbalance.  Say it happened to be positively charged.  Now positive charges in the region are more strongly repelled, and negative charges are more strongly attracted.  This imbalance tends to drive the net charge on the object toward neutral.
