Is there a type of surgery that could make my jaw more forward than downward? Would it make my face shorter?

You do have a "weak" chin, but if your teeth don't bother you (they fit well together) and you don't want a very long surgery and recovery, a chin implant will help considerably. This can be done in the office, through the mouth so there are no visible scars.  They work great!

Maxillofacial surgery is performed to correct growth problems with jaw that affects the teeth and bite. This surgery usually leads to a dramatic improvement in facial balance. If you a normal bite then a genioplasty procedure can be performed to lengthen your chin. I would recommend a consultation with an oral & maxillofacial surgeon to determine if you are a candidate for this surgery. Good luck

Hello dear
Thanks for the question and provided information as well. 
My recommendation for you is a personal consultation with a certified plastic surgeon to be able to make a complete evaluation and suggest the procedures that can fulfill your expectations and provide you with the best results.
Good luck :)

The first step int your quest to improve your facial shape is to get an orthodontic evaluation and see if you are a candidate for orthognathic surgery. Your lower jaw seems short and your bite may be off as well. If you are not a candidate for major jaw surgery then a sliding genioplasty to bring your chin forward and up can be done.

Your "assessment " appears reasonable so visit an orthodontist and perhaps a Maxillofacial surgeon for the assessment of an expert and they will guide you accordingly. The process is detailed but common so you can be comfortable about improving the bite and jaw structure