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.
A swab taken from the tip of the penis is spread on a glass slide, stained and examined under the microscope for signs of infection. This test can only be done at a specialised sexual health clinic. The result is available immediately. This test will also detect gonorrhoea in most men.