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When some materials are chafed one to another, they obtain opposite electric charges. Does it mean that if these materials will then be connected to the ground, the direct currents of opposite direction will occur? And if we put some dc device like a lamp in the open of wire connecting statically charged material with the ground, should it light until the static charge is gone?

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You're quite right that Static attraction is produced by friction. Because while rubbing, On thing to notice is that the name "static" comes here because the charges are not moving in the conductor (or insulator). Only a moving charge could produce current and hence the name "current" electricity...

Current is defined as the rate of motion of charges across any cross-sectional area of a conductor. I think it doesn't apply to static charges.

Note: Static electricity could be produced along with currents. A much higher differentiation is that Static electricity could cause extreme voltages (Lightning, Van De Graaff generator) which implies a very low current (to flow).

UPDATE based on comment: "When the voltage becomes large enough, the current resulting is Lightning". As I'm quite low in English-language skills, I assume that there are two possible ways -

(1) A typical lightning lasts from several milliseconds to 0.2 seconds which is very low for our eye to perceive and resolve... According to Wiki,

An average bolt of negative lightning carries an electric current of 30,000 amperes (30 kA), and transfers 15 coulombs of electric charge and 500 megajoules of energy.

So, you need something like hundreds of yards of powerful batteries or capacitors to store such a massive energy. But, it's somewhat difficult (I think it's impossible) to capture such a large amount of energy within such a short period of time.

This link would be useful for better explanation...

(2) As you mention something using a long conductor, I think it's based on Lightning conductors which attract the charge of clouds to a high metal rod (at ground) that redirects and dispels the current underground. In a lightning rod, opposite charges are induced by static induction and this repels the like charges in air, thereby lowering the potential of cloud. But, The current flow in the rod is also small somehow... 'cause the rate at which the charges induced would be proportional to the rate at which it's discharged (or neutralized)...

(Instead of a neutral conductor) If you place a conductor with same charge as that of the cloud (bottom layer), the same thing happens as mentioned in (1) and you'd obtain the lightning directly upon the rod and the rod would probably melt..!

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Yes, so that's what I am asking about. The lightning happens because there is the difference of potentials between the ground and clouds. And when this difference becomes large enough, the current occurs between them resulting in lightning. Consequently, I suppose, that if there was some conductor between clouds and ground, so that it's conductivity is much higher than one of the air, then the clouds should discharge with much lower voltages during longer period of time. So probably the current would still be very low, but voltage would be much smaller. Does it make sense for you? –  BartoNaz Oct 17 '12 at 15:11
    
@BartoNaz: Hello BartoNaz, I've revised it based upon your comment. Re your last sentence: "the current would still be very low, but voltage would be much smaller". How would both the terms be small (as they're inversely $\alpha$ to each other). Also, I was wondering how longer the cloud would stay there (at the same place) till you complete extracting its charge through a small rod... :-) –  Waffle's Crazy Peanut Oct 17 '12 at 17:18
    
Well, I don't see any inconsistency with low current and voltage, since the discharge of the cloud will be distributed over longer time then in case of regular lightning. And the resistance will much lower with the conductor. But I am asking about theoretical principles. Of course I'm not going to attach the metal rod to the clouds. I am thinking about a general principle of collecting and storing electrostatic charge. –  BartoNaz Oct 17 '12 at 20:56
    
I believe that whenever you have an object with static electric charge on it and an electrically neutral object , after connecting them by a conductor the flow of electrons should occur from charged object to neutral one. Like it is in the rechargeable battery, where you have two parts with different concentrations of electrons, which results in flow of electron from part to another when connected, until the balance of charges is established. But maybe I'm missing something. –  BartoNaz Oct 17 '12 at 21:06
    
@BartoNaz: Well, Sorry BartoNaz 'cause I can't give any kind of theoriez for that..! BTW, I didn't ask you to do some home-experiment. I just suggested that "This is what happens" (The fraction of energy you'd obtain from Lightning would be too low). Hence, the cloud would move away... But - It's a good idea though :-) –  Waffle's Crazy Peanut Oct 18 '12 at 2:17
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