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.
Patient : I recently was with a girl. We didn't have sex, but my penis was rubbing against her vagina, she was wearing underwear. A week later she tells me she has genital warts. Could I contract HPV through this interaction? I didn't touch her vagina directly, but cloth is only so thick.