I managed to stop the bleeding with flour, confine her with a chew toy so she doesn't aggravate it, and shower the worst of it out of my hair.
Quick question, because I need to go to work: Say in all this mess I got dog blood in my eyes and mouth. Is this bad? Do I call a vet? A human doctor?
Edit: Called vet to ensure that dog is not going to bleed out. They say she might need a suture but we can wait and see if the bleeding stays stopped as long as she has energy and isn't showing signs of imminent death. However, the Crazy Chinese Dog Disease is very transmissible to humans via blood, apparently. The internet disagrees on that front and says it can only be contracted directly by a tick, but I should probably listen to the licensed medical professional who says "Um, your dog scratched open wounds in your hands and you were covered in parasite-filled dog blood? CALL YOUR DOCTOR."
Edit2: Doctor sez: "URGENT CARE 4 U!" I says, "I'm at work! Will my eyes explode if I wait until tonight?" He says, "Prolly not, no." So anyway, I'm sure this is all no big deal, but I'll go in tonight and hopefully they'll stick me on some preventative antiparasite stuff or whatever and then I can go BLEACH EVERYTHING I OWN.