Is doxycycline x4m normally prescribed for a 12 yr. old boy who has mild to severe acne? doesn't prolong use of abx cause immunity/resistance later?

Probably fine: Often the first line of treatment for adolescent acne is topical, including topical antibiotics. If your doctor feels that oral antibiotics are needed, that is a judgment call based on experience. Doxycycline is not an antibiotic that is used to treat the usually acquired infections, but can be a good choice for acne. Talk to your doctor about your concerns, but 4 months is ok , if working

Yes and no: This is a pretty standard approach to a kid who has reached a stage where oral meds are considered. Some docs use this continuously and some will alternate this med with another, month on/off for both to prevent emergence of resistant strains. This treatment is never done in a vacuum & I suspect the doc has scheduled follow up to decide when /if modifications to the plan are needed.