2. Louis Pasteur proposed the germ theory of disease in the late 19th century. People in the Middle Ages did not understand germs, but they did have some conception of disease, infection, and transmission. For the most part, it was a rudimentary idea that exposure to "bad air," which they called miasma, could cause infection. They also believed that diseases could be caused by poisons (see well-poisoning accusations from the plague), and had a basic understanding of the dangers of rot and excrement (but not actual knowledge of bacteria). This meant that they were fully aware of the danger of disease from things like corpses and dung, just not necessarily why.

No, English archers did not cover their arrows in shit or other substances with the intention of causing infection. While sticking their arrows in the dirt may have led to some wounds getting infected, this was more of a happy accident than a deliberate policy. Even if it did happen, it wouldn't have much of an impact on the immediate battlefield. If an enemy wasn't outright incapacitated by the arrow wound itself, it's not as if infection would instantly set in and cripple him. Besides, when English archers wanted to make sure that a wounded Frenchman stayed down, they would just slit his throat or bash the guy's head in while he was crawling around.