Non-zero net charge on cosmological scale It seems very clear, that because atoms and ion compounds are electrically neutral, our world seems to consist of an equal amount of positive and negative charge resulting in a zero net charge.
What would be the consequence of (small) non-zero net charges on cosmological scales?
Planetary
What would happen, if we gave an otherwise electrically neutral planet (e.g. Earth) 1000 (1 million, $10^{23}$, $10^{30}$,$10^{50}$) addtional electrons?
Sure, these addtional electrons would somehow diffuse. Could we even measure this diffused charge or at least proof, that there is an additional charge?
Galactical
Could a non-zero net charge in (parts of) galaxies explain their behaviour (e.g. rotational velocity, expansion) better than dark matter and dark energy?
Suppose one galaxy had a positive net charge and a neighbour galaxy a negative net charge. How long would it take for these charges to neutralize itself?
 A: I'm not sure what kind of answer you expect to this question.
First, let's look at planets/stars:
For "small" charges (a few thousand electrons on a planet), nothing happens. It is just a weakly charged body strutting through space.
For "big" charges, you get a body that will, at least if it is an an orbit or an otherwise accelerated motion, continually radiate significant amounts of energy in the form of bremsstrahlung, which will sooner or later make the planet lose kinetic energy and thus destabilize its orbit.
Of course, excess charge will generate electrical forces acting upon other charged bodies, and if all your planets are equally charged, it will look just like they repel each other and thus weaken their gravity. If the EM force exceeds gravity, this will break up the "bound" gravitational system and make it impossible for these charged bodies to form a stable system.
Galaxies are not different. They are just bigger, and the charge has to sit on their individual constituents, anyway. 
The part of your question where you ask about the charges "neutralizing" seems to indicate that your picture of electromagnetism is a bit unclear: An exchange of charge between two bodies holding opposite charges can only occur if there is something between them that can conduct the charge - for galaxies, there is only vacuum (or rather, very thin neutral interstellar gas) between them, and it cannot transport charge in the general case, so they will never neutralize until they "touch" each other, i.e are close enough that for example two of their charged planets are close enough that their atmospheres touch and the charge (which has to be very big for this to occur) ionizes the atmosphere.
Also, charged planets are not as easy to build as you might think - planets form out of clouds by very weak gravitational attraction (and even this is not fully understood), and the presence of a large amount of charge would drive the constituents stronger apart than their gravity pulls them together - overall electrical neutrality is a requirement for planets to be formed by gravity, which is rather weak in comparison.
