Should I get my healthy premolars extracted for braces?

You need to have some teeth removed in order to eliminate both your crowding and your protruding teeth.  Without extractions, your teeth will stick out more than they do now.  What you read on the web about extractions is wildly inaccurate.  In general, about 20-25% of all patients seeking treatment need one or more teeth out to correct their smiles and bites.  You are one of those.  If you want to see how unhappy people are when they don't get teeth out when they should have, search Realself for "Unhappy after braces" or "teeth stick out after braces" or "overbite left after braces" to see the other side of this debate.  That said, you should have been told about extractions before you had the braces attached.  You also mentioned dentist and not orthodontist.  Extraction cases are more difficult than non extraction, I hope you are being treated by an orthodontic specialist.A couple of other points:  Orthodontic treatment can't make your mouth smaller (size is determined by the jaw bones, not the teeth).  "Jaw development" is not a term that has any meaning in orthodontics, and orthodontics of any sort does not cause TMJ issues  (hundreds of studies have been done).  When you search this site regarding braces, you will see that when an orthodontist answers, they are careful, well thought out answers.  They never attempt to scare the patient away from any treatment using inflammatory terms and false statements.

Extracting the premolars can lead to a number of harmful repercussions including gingival recession, abfraction of the root, and most importantly TMJ issues.  Essentially, "retraction orthodontics" can lead to airway issues that can affect sleep, tongue position, and the ability to breath through the nose.  
This is a topic that can be debated within the profession with great passion.  With my experience in TMJ treatment with a physiological approach, I would strongly suggest you get a second and third opinion.  Premolar extraction is not something I would recommend. Period. 

Hmm, tricky question. Your overbite is very possibly caused by damage to your jaw joint as a child. This expresses in deep overbites. Dr. Mark Piper of St. Petersburg, FL. gives an excellent lecture on the subject. The cause is one concern, the treatment is entirely different. You aren't going to attempt to repair the  joint, and that probably cannot be done anyway. So getting down to your question of treating the symptoms of excessive overjet/overbite, premolars can be extracted which will be the fast way to treat. Another option would be to distalize the upper teeth two at a time. This takes longer but preserves the premolars. Either way you need to find and trust a dentist or orthodontist with plenty of experience because your treatment will not be one for a novice. Good Luck. Hope I helped and don't put a lot of belief in what you read on the internet.

The underlying problem in your case is that your upper jaw or maxilla is TOO SMALL!  Taking out 4 teeth to do orthodontics will make your mouth smaller and crowd your tongue.  If you want a full wide healthy smile with good tissue support and want to eliminate the dark buccal corridors you need to 3 dimensionally develop your maxiila. Four on the Floor Extraction/ amputatiion is definitely not the best approach.Look into the DNA Appliance and Epigenetic orthodontics / Orthopedics.  The process will grow you a larger healthier nasopharyngeal airway, this process has been called pneumopedics.A SARPE procedure is possible and it was routinely done before the DNA Appliance. I prefer to avoid surgery in general but it is preferable to amputations.  There are other types of expansion such as ALF but in a 19Y/O the DNA Appliance is my first choice.Ira L Shapira DDS, D,ABDSM, D,AAPM, FICCMOPast Chair, Alliance of TMD OrganizationsDiplomat,  Academy of Integrative Pain ManagementDiplomate, American Board of Dental Sleep MedicineDiplomate, American Board Sleep and BreathingRegent, Master & Fellow, International College of CranioMandibular OrthopedicsBoard Eligible, American Academy of CranioFacial PainDental Section Editor, Sleep & Health JournalMember CRANIO Editorial Board, Journal of Craniomandibular and Sleep PracticeMember, American Equilibration SocietyMember, Academy of Applied Myofunctional Sciences Member, Academy of Cosmetic DentistryLife Member, American Dental Association