How do you deal with worn wires?
These were both breaking right where the hinge moves, so I was wondering, is this a common problem on all moving-wire electronics? 


It becomes a problem if it is not handled correctly by the design. Non moving cable parts should be fixated so they don't get worn by vibrations. The wear can occur on the insulation or at the joints, at a connector or PCB. Moving cable parts should be allowed to move freely (within the cables specified allowed bending radius) or be protected by some means. Flexible pcb's, cable carriers, cable chains and reinforced insulation are common techniques.  


  One cable seems to have cloth-type covering over it, and I didn't see any damage to it - what is this covering called, and can I add some to the wearing cable to prevent further damage? 


It is probably braided cable sleeving. Can be made of fiberglass, cloth or even steel. Check for "wraparound" sleeving if you want to add it to cables which aren't possible to detach.


  Or would a spot of silicone plastic sealant work as well?


The important part is to understand what is causing the wear (heat, friction etc. ) and what kind of damage that has already occurred. If the insulation is broken on several conductors close to eachother, it is usually a good thing to take the conductors apart and reinsulate them individually.