RE: can you repair carbon arrows
If it's badly damaged, I chop it shorter, refletch, and give it to a buddy with a shorter DL.
If it's just slightly split (1/2" or less), I remove the nock, smear the end with epoxy, wrap it tightly with serving, smear a thin coat of epoxy over the serving, let it cure, and then reinstall the nock. I've done this many times with no problem (heck, it's stronger than the original end), and they fly fine.
It's the first arrow I use for steel does or plinking; that way I don't feel as bad about losing it.