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.
STDs do not always cause symptoms, so it is possible to have an infection and not know it. That is why it is important to get tested if you are having sex. If you are diagnosed with an STD, know that all can be treated with medicine and some can be cured entirely. There are dozens of STDs. Some STDs, such as syphilis, gonorrhea, and chlamydia, are spread mainly by sexual contact. Other diseases, including Zika and Ebola, can be spread sexually but are more often spread through ways other than sex.