My third hand damages wires insulation because it's holding them too tight. Any tips how to avoid this?
Wrap the teeth in heat-shrink tubing or electrical tape. It will still hold just as tightly, but without the teeth, the pressure on the wire being held will be less.
I like to tightly wrap rubber bands around the back of the clips to take some of the pressure off. That way, you can adjust the pressure to your liking. Sometimes I use the alligator clips to hold a pair of cheap soldering heat sink clips and use those to actually hold more delicate components.
See the heat sink clip in this picture:
I have the same tool; I just bent the teeth on the clips inwards using a pair of pliers.
You could also change out the alligator clips for a spring clip instead. I found some small clips similar to this at a hardware store (http://www.homedepot.com/p/HDX-3-4-in-Mini-Spring-Clamp-80001/100656782?MERCH=RV-_-RV_nav_plp_rr-1--NA--100656782-_-N). They still squish the insulation a bit, but at least they don't bite into it like an alligator clip does.