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?
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,
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..!