> Is it possible to provide power to signal repeaters with POE to get around the ~100 meter limit?
Yes, but at that point you may as well just run fiber. These days, the SFP modules are cheap (20km SFP for ~$80).
The loss of signal on a CAT6 cable will be much greater then on fiber. When too much noise is introduced on a CAT6 cable, speeds will drop considerably. Packet loss will be quite high.