The right cartilage of my nose seems to have either caved in or flipped. What are my options with out ruining it worse? (photo)

Cartilages do not flip. You are likely to have had some scarring from ingrown hair infection. It is not possible to pinch nerves like you have described. Self diagnosis is a dangerous thing. See a plastic surgeon for any options.  A personal exam of your nose is the only way to give you the answer that counts.

Hello asmi1992,An in person consultation is really the best option.  The reason is it is important to determine if what you are seeing is the underlying cartilage or if that is scar tissue from the bad infection, as the treatment options are different.  Because of the location of the cartilage and nerves, it would not be likely that the cartilage is pinching a nerve, but it could be the area is uncomfortable from scar tissue or other causes of pressure.  Again, the best way to evaluate is in person, so I'd recommend you consult with a facial plastic surgeon. I hope this helps and good luck.