breathe contaminated air in the general vicinity of animals, drink water they've pissed or shit in, etc.

Don't go near animals. Boil water before drinking. Simple as.

That is in fact how a vaccine/our immune system works, yes.

Vaccines work using either diluted toxins, or increasingly, proteins / RNA that just look somewhat like the real thing. The immune system works by enough of the population dying off until only those with the necessary mutations are left.