I know that insulators do not conduct electricity because they do not allow the free movement of electrons. Let's assume the object that's going to touch the insulator is negatively charged. Does some charge get transferred to the the insulator or does no charge get transferred at all? I want to understand this more conceptually. I don't really get why insulators are used in electrostatic experiments.
When a charged object touches an insulator there is usually no charge being transferred unless the field strength is so high that you get corona discharge.
For example, look at the electrophorus in this video. It shows that you can put a metal disk onto an insulating plate without any charge jumping over. At the end of this video you can also see another example where charge is being transferred – in this case between aluminum foil and a plastic cup. That happens because the field strength just got too high leading to corona discharge through the air.
There is a great explanation of that charge transfer in this video starting at 33:00 he explains the process in detail.
Actually, it is most likely that the charged object is an insulator.
Conductors are very hard to charge. Any charge you put on them disperses through the surface, so the charge density is pretty small and there is very little repulsion (and the field given off will be small). Touching a charged metal ball will not give that much of a static shock because there is no strong impetus for the charges to leave the ball.
On the other hand, on an insulator, the charges can only spread on a limited portion of the surface. This increases charge density greatly, and there is a lot of repulsion and thus stored energy. The electrons, being in a repulsive field (with high potential energy), will take a route off the insulator if provided. So charges on an insulator are more mobile when it comes to leaving the insulator, and thus we use insulators in electrostatic experiments. Metals "eat" charge and don't discharge as much, insulators "hold" them until they have a chance to discharge.
Well if you don't go into the micro details where the charged sphere may induce extremely low quantity of charge in the insulator before touching and then transfer equally less charge even on touching which would not even be practically noted, NO! There is no charge transfer.
Conceptually you can say the extremely high resistance of the insulators does not give the charge carriers a tempting path and place to go to. As we know that charge normally travels over the paths which offer lower resistance.
For your next question, it would be wrong to say why insulators are used in electrostatics as it sounds like thats onky where they are used, insulators play an important role in electrodynamics (current electricity) just as well but that is not your question. Insulators have a special property, unlike conductors, if you charge an insulator, just by touching it at a particular point with a charged substance (most probably rubbing with something) the charge stays localised, had it been a conductor the charge would have spread over its entire surface and we would observe lesser effects. But since in insulators the charge stays localised, it shows significant effects, which are easily noted.