Why can insulators be charged by friction but not by conduction? When you charge an insulator by friction, the close contact of two electron clouds (each from a different medium) allows for electrons to be transferred between them. Yet why can the same not be accomplished with conduction?
 A: 
When you charge an insulator by friction, the close contact of two electron clouds (each from a different medium) allows for electrons to be transferred between them.

The number of atoms coming in contact to each other by friction is proportional to the area of the two insulators rubbed together. The whole surfaces can be rubbed . That is an enormous number of atoms.

Yet why can the same not be accomplished with conduction?

Suppose one has a battery, and wire attached to one of the poles. If one touches a metal plate, it will be charged because metal conducts well and all the surface will come into balance and display the voltage of the battery. On an equivalent insulator plate only where the wire touches the surface the charge will appear,  a few atoms, depending on the area of the point of the wire,as by definition the mobility of electrons is very low in insulators.
A: Are they insulators? They do not conduct charges very well by definition,
and thus one should choose another approach instead of trying the 'conduction' way.  
A: It can be done. Just spray electrons on an insulator, they will stick.
This is for example used with paint drops for painting metals. The paint drops are charged by corona ionizing the air near the nozzle and the piece of metal to be painted is charged, this way less paint is lost (as the paint drops are attracted to the piece of metal).
Furthermore, at high enough electrical fields everything will conduct.
So it is not impossible, it is just difficult to uniformly charge the surface of an insulator with a single stationary wire (as the charges will barely travel through the insulator), but if you brush above the insulator with many wires at high voltage and you can easily charge it by conduction. Why high voltage? Because the surface will have a very small capacitance, so high voltage is necessary to deposit noticeable amounts of charge.
