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.
STIs often have no signs or symptoms (asymptomatic). Even with no symptoms, however, you can pass the infection to your sex partners. So it's important to use protection, such as a condom, during sex. And visit your doctor regularly for STI screening, so you can identify and treat an infection before you can pass it on.