The distinction between organic and inorganic carbon compounds, while useful in organizing the vast subject of chemistry... is somewhat arbitrary. Organic chemistry is the science concerned with all aspects of organic compounds. Organic synthesis is the methodology of their preparation.
No, it is not possible to contract the typical STDs, such as syphilis, gonorrhea, chlamydia, hepatitis B, etc. from wearing someone else's dirty underwear. However crabs and scabies, may be spread in this fashion.
1 After using the toilet, wipe from front to back to avoid spreading yeast or bacteria from your anus to the vagina or urinary tract.  Wear underwear that helps keep your genital area dry and doesn't hold in warmth and moisture. One good choice is cotton underwear.