I'm not aware of anesthesia increasing or decreasing hair growth. Some patients have some head hair fall out from anesthesia, but this is rare and usually fairly limited.
Local anesthesia should be even less of a problem.
There's a chance that your hormone levels are changing (which happens to many teens). This can cause additional hair growth and may not be related to your surgery.
I'm sorry to hear about the problems you had with your nose surgery. You can complain to the state board of medicine if you feel your doctor acted unethically or was not competent.
If your results left you disfigured (where someone who didn't know you would think that you had a botched nose job without even telling them), you might be able to talk him into a corrective surgery or even sue him if it's bad enough. Both these options will take time and money, though.
You will also need to prove gross incompetence, which is a very serious charge stemming from a serious breech of conduct. Usually, mildly unsatisfactory cosmetic results will not be covered by this. New York State: Office of Professional Misconduct
If you think you have a case for malpractice, etc., you should contact a lawyer.