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"And Moses made a serpent of brass, and put it upon a pole, and it came to pass, that if a serpent had bitten any man, when he beheld the serpent of brass, he lived." - Numbers 21.9
Afaik, thats actually not supposed to represent snakes, thats a medieval interpretation. Its supposed to represent the stick used to get some kind of worm infestation out of the body, by "spinning" the worm slowly onto the stick.
No, that is just a myth. Most likely the symbol comes from Asclepius.
The parasite you mean is /Dracunculiasis, but that is not where the symbol comes from.