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.
Keeping the skin cool, dry and clean is by far the best way to prevent reinfection of fungus infections (unless you have a medical condition or are taking medication that has side effects of fungus infections). Wearing breathable, anti-bacterial underwear helps keep your skin cool, dry and clean.