Hello,
Boils are caused by infection of the hair follicle by staph aureus. Hair follicles that are blocked by greasy creams, petroleum jelly, or similar products are more vulnerable to infection. Bacterial skin infections can be spread by shared cosmetics or washcloths, close human contact, or by contact with pus from a boil or carbuncle.
Apply warm compresses and keep the area clean. Drainage is done only when it becomes soft and forms a head. Take some over the counter pain reliever and consult a doctor for examination and antibiotics prescription. If there are recurrent boils, then get your blood sugar levels estimated.
Regarding the wart, it can be confirmed by examination and biopsy skin.
It is very difficult to precisely confirm a diagnosis without examination and investigations and the answer is based on the medical information provided. For exact diagnosis, you are requested to consult your dermatologist. I sincerely hope that helps. Take care and please do keep me posted on how you are doing.
Possibly a folliculitis and your family doc can do the needful.