Natural Healing. The good news is that area has fantastic blood supply and so healing will occur rapidly. Inflammation (red, hot, tender swelling) is the body increasing the blood flow to the injury to bring healing factors to the area and also the veins expand to carry bad stuff away. Dressings and ointments are of no value there. Avoid further trauma to the area and let nature take it's course.
Yes. The neosporin won't hurt but it may not help. If you are having burning and itching, it may be a yeast infection. There are a bunch if over the counter medications you can use. Adding some Hydrocortisone cream to the area will also help. If it doesn't get better soon, you should see your doctor to make sure it isn't anything else.
Unsure. I have never seen, nor heard, of using neosporin inside the vagina. It may not be harmful, but it may not be helpful either. Better to know just what is causing your symptoms and treat appropriately, which will likely require you seeing your doctor for an exam. If you are asking about using neosporin on the skin outside the vagina - it would be as safe as using neosporin on any other skin.