Will toothpaste help bring my boil to a head?

Use warm compresses. Not sure about the toothpaste, although to be honest i would avoid it to prevent irritant contact dermatitis in that sensitive area. Warm compresses will help express the contents. Avoid squeezing the area as well and jf you see streaks or surrounding areas of redness you should see a doctor because you might need antibiotics.
No. You need to see a doctor for that, you can clean the area with peroxide or alcohol in the interim.
Infection. Sounds like you have an infection. Toothpaste will not get rid of it. Find out where the infection is coming from and get it treated. See a dentist!