My question is quite simple I think. Is it true that we have approximately the same number of electrons as we have protons and the average electric charge of the molecules in our planet is roughly neutral?
We can see that the charges in the observable world are close to balance by considering what we would observe if they were not.
Big imbalances of charges cause strong electrical fields. Those are measurable and their effects are observable. We can create such imbalances locally (that's what a Van de Graaff generator does) and the effects are spectacular: really big sparks as the strong fields ionise air and cause big electrical discharges. These separations also occur in nature (that's what thunderstorms are). In both cases, the consequence of significant temporary separations of charge are spectacular natural phenomena that act to restore the balance.
It could be true, however, that the earth as a whole had a net charge. We wouldn't notice this while on the surface as the net charge would be relative to interplanetary space. On the other hand any such charge would probably be neutralised by selective interaction with the charged particles in the solar wind which we could observe. Also, if the net charge were substantial, we would observe significant effects on planetary orbits (the electrical field is a lot stronger than the gravitational field.
These observations together suggest that both locally and globally we have approximately the same number of protons as electrons on the earth.