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.
Underwear can exacerbate problems of vaginal dryness. Making sure underwear is an appropriate cut and made from natural fibers can help ease vaginal dryness, whilst not interfering with any other treatment options a woman may follow.