One side of my nose is flat/caved in. What would you recommend? (photos)

The nasal areas that you described and
would like improved can be addressed with an artistically performed nasal surgery (rhinoplasty) and possible septoplasty.Keep in mind that following the advice from
a surgeon on this or any other website who proposes to tell you exactly what to do based
on a limited two dimensional photo without examining you, physically feeling
the tissue, assessing your desired outcome, taking a full medical history, and
discussing the pros and cons of each operative procedure would not
be in your best interest. I would suggest you find a plastic surgeon certified
by the American Board of Plastic Surgery and ideally a member of the American
Society for Aesthetic Plastic Surgery (ASAPS) or facial plastic surgeon (otolaryngologist) with extensive experience in performing cosmetic nasal surgery that
you trust and are comfortable with. You should discuss your concerns with that
surgeon in person.Robert Singer, MD FACSLa Jolla, California

It appears from your photo that your lower lateral cartilages that produce the structure and shape to your nasal tip are misshapen. The right side is more of a concave shape which is producing the shadowing and depression on that side.  It is difficult to say if the piercing contributed to this. Regardless, this is treatable and involves using cartilage grafts that would be harvested from the nasal septum and placed with the lower lateral cartilages to create a more symmetric appearance. 

The  photograph demonstrates an inverted  lower lateral cartilage that is concave, rather than convex like the opposite side. A rhinoplasty procedure can address making the changes necessary to make the nose more symmetrical. This would involve harvesting a cartilage graft from inside the nose and placing the graft in the concave area to match the opposite side of the nasal tip. For many examples of rhinoplasty, please see the link and video below