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.
Gardnerella is not a sexually transmitted disease, as men cannot spread the bacteria that causes gardnerella to female partners. However, female sexual partners can pass the infection to each other through shared sex toys and vaginal secretions.