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.
It is not a sexually transmitted infection but can develop after you have had sex. Men do not get bacterial vaginosis. This page gives you information about thrush and bacterial vaginosis, what you can do if you are worried that you might have either or both infections and how to get treatment.