Can a primary care doctor physically treat acne? (like by extracting the nodule instead of prescribing meds)

Probably not: There's no law against it, but most primary care doctors have little training and experience in doing that.

Depends: Each PCP may have different areas of focus or experience. You should inquire if that is something they routinely do in their specific practice. Acne usually requires a multi-faceted approach - medication is an important component but chemical peels and other treatments can be very helpful. Sometimes an underlying disorder may be the cause of acne so it is always good to have an exam.