Not a doc, just a guy who has gotten in his fair share of fights in his younger years and traveled some drier climates as well. A stiff bit of folded up paper held between your upper lip and gums, held as tight as you can stand it, will help stifle the nose bleeds when they happen.
For the dryness, try swabbing the insides of your nose with Vaseline using a q-tip. It'll help keep the skin in there from drying out too bad. It'll also help the wounds to heal. Neosporin will also work as well for this.