Can exposure to blood make pathogens resistant to the immune system?
You have got the concept of vaccination a bit wrong. Pathogens have to evolve to acquire resistance to immune system (or antibiotics). Technically, the white blood cells also evolve to acquire resistance against pathogens if they don't have a strong innate immunity. Development of immunity after vaccination is also not instantaneous. The pathogen population has to "grow" in the blood for at least a few generations to become resistant. A drop of blood would probably not give the pathogen enough time to evolve. Moreover, your white blood cells will die quickly outside the body unless they are cultured in a controlled environment. Therefore, the selection pressure to evolve would also not be there.