Do I really need to be anointed with oil?
Yes. There are two types of oil used in baptism: olive oil, which is either poured into the font water (Russian tradition), or applied all over the body (Greek tradition, when babies and tots are baptised). Check with your priest how he does it.
The other oil is chrism, which is applied in the form of a cross on the forehead, lips, ears, chest, hands and feet. Chrismation is not negotiable. No chrism, no baptism.
The second question is about having my hair cut in four places to represent the Cross. Is this really essential because I'm not sure if my parents will allow that.
Again, not negotiable. Tonsuring represents "entering the fold", a small sacrifice. The amount of hair that is clipped off is no more than a few strands at each clip. Practically or completely invisible.
but oil causes acne
Not at all true. Complete myth. Teenage hormones cause acne. And you're a teenager.
