how to extract Glycyrrhizin from liquorice root?
Assuming that you have Glycyrrhiza glabra, the glycyrrhizin content should be somewhere less than 25% of the content - which is plenty for an extraction, and means you won't need a whole bunch of root and specialist techniques to extract it.

A quick google search leads me to believe that the procedures for extraction of the glycyrrhizin are fairly simple - maceration (literally chewing, but in this context meaning mashing/grinding/blending) followed by steeping for about 4 hours at room temperature (see also here).  It looks like water alone is a pretty good solvent for it, but a mixture of 10% ethanol and 90% water is better.

From Wikipedia, it looks like the solubility is about 10 mg/ml (milligram/milliliter) in water. This is assuming that you can get a saturated solution from a pure compound. You will likely not get this sort of concentration, at least not without a bunch of other compounds coming across and giving you a strong liquorice flavour. Note that the average detection limit for sugar/sucrose is about 12 millimol/litre or about 4 mg/ml by my calculation - so with something 50x sweeter, you should have no problem sweetening things to a similar level as sugar using a solution with as little as 0.08 mg/ml. Your major problem with attempting this process is going to be how to measure how sweet your extracts are - you can do it by taste, but that will be very subjective. 

Also note that many recipes for things like cookies call for crystalline sugar to provide structure in the batter rather than purely for sweetness, so you will need to substitute some of the liquid for your liquid sweetener and work out something to add to maintain the structure. I am sure that there this is a well worked out solution for this already for people wanting cookies without the sugar, so I will leave it to you to research.