There are some problems with how your question is phrased. Can we induce charges on things? Yes. The electrons of something, though, must go someplace, which would mean that a planet with a charge would require removing those electrons.
Then there is the problem of Coulomb forces; the force of attraction between positive and negative charges. Coulomb forces are much, much stronger than gravity. This can be easily done by looking at the equations of attraction for charges
$$F = \frac{k_eq_1q_2}{r^{2}}$$
and masses:
$$F = \frac{Gm_1m_2}{r^{2}}$$
(These are scalar equations, but the vector versions are not too different, and do not meaningly impact the discussion here.) If we look at some protons at 1 m, the values of each of these forces is $2*10^{-28} N$ for the electric force and $1.87 * 10^{-64} N$ for the gravitational. (For electrons, it's a gravitational force of $5.54*10^{-71} N$ and $2.3 * 10^{-28} N$ for the coulomb force at 1 m.)
I hope you see the problems here. The gravitational force is simply too weak to hold things which the electromagnetic force wants to pull apart. You cannot assemble a planet from a plasma because of this.
There is also the issue of measuring charge. We always measure charge according to a base level of charge, usually that of the earth. If we had a solar-system spanning network of wires, maybe we would see some electron flow from one planet to the next. I find this unlikely, though, because any discrepancy in charge in the universe will attempt to fix itself quickly via Coulomb Forces.
Also, electric currents we carry along in power lines are shoving electrons back and forth, in an alternating current. Modern power plants do not package electrons and move them somewhere to power our devices. Even in direct current situations, the electrons which move in circuits are replaced by electrons directly behind them. The closest thing you would get is a capacitor, and those discharge rather quickly.
Now to your questions, directly.
Can We Charge a Planet?
Yes. We can theoretically remove electrons from planetary bodies. No, it is not practical. You need to gather your electrons, take them away from the planet, and put them on something which will keep them there. Holding electrons on one thing is hard. If you get a big enough bucket of electrons (no, you can't place them in a bucket like apples), they will overcome the resistance of your bucket and go back to where they were.
Stopping other electrons in the universe from taking their place is harder. How are you going to stop space dust from hitting your planet and evening out the electric charge? Even something like the solar wind will stymie your plans by introducing ions of the opposite charge. Say you remove some portion of a planet's electrons, but more matter is added from impacts of things in space. Sure, you're losing electrons, but you may be gaining more electrons than you can pull away. This would result in you shipping a lot of electrons, but your shipped electrons would be an ever-decreasing percentage of the total electrons.
If you remove enough electrons from a planet, chemistry will be utterly displeased, and you will see compounds break down. This is due to the fact that you need electrons to form chemical bonds. When there are not enough electrons to go around, expect everything to dissolve in a highly acidic (according to Lewis) mess.
What About a Charged Earth/Moon System?
If electrons from earth ended up on the moon, in such a way that the moon had an opposite charge from that of earth, their orbits would change. We have established above that the EM force is much, much stronger than gravity over the similar distances. The difference in charge would lead the charged particles on both the earth and moon to move towards each other. If those particles are affixed strongly enough to either of these bodies, it will carry the body with them.
Can a Planetary Body be Charged?
In theory, yes, there could be situations where a planetary body has a net charge. This situation may not last very long on a cosmic timescale, due to the attraction of charged particles to particles of unlike charge. Perhaps something is emitting a large stream of super electronegative particles, and they are ripping electrons off of a planet. This can't be a regular solar wind, as that is made of both positive and negative ions. (The ions of opposite charge could neutralize your charged planetary body, resulting in a charged solar wind or odd topographic changes to the wind.)
Perhaps there is a net charge of the universe in general, maybe due to excessive production of particles of a certain charge. We wouldn't notice, of course, unless something deviated from that. This is because we do not measure how many protons and electrons are in a thing when we measure charge, we only measure how many more or less of those there are relative to something else.
Transferring Power?
You mean transferring energy? Power is not energy. Really, it ought to be transferring electrons. Anyways, if you charged a planet like a giant Van de Graaff Generator, you would find that whatever comes into contact with that planet will try to get into electrostatic equilibrium; that is, it would transfer charge between the new object and itself until they both had that same overall charge. That makes a lot of lightning.
You may also get an effect where objects of like charge (from a solar wind, for example), would be repelled by the planet, whereas it would "suck in" items of opposite charge. It would be a rather confusing thing to see until you realized the charge of the planet.
