Yep, did this before and it really hurts. I don't think an infection can take hold so quickly, but your nerves are probably reacting to the cut. Wash it out with soap and water, maybe use a little triple antibiotic cream if you're nervous about infection, and keep it clean and dry. If it doesn't stop bleeding, you'll need to get it looked at by a medical professional.
Not as yet. You might want to get medical attention if it is a deep cut. The tingling is most likely because you cut a nerve. Keep the cut clean and covered with a sterile bandage and you should be ok.