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.
If you are going to wear underwear when working out, look for fabrics that are non-abrasive and that have flat seams. Men should wear boxer briefs for sports with medium intensity and jockstraps for high intensity sports.