NO. It cannot.
Fart on pillow eyein. No most of the flatulece contain bad gases like methane and others and will not cause pink eyr (conjunctivitis).
No. The pink eye virus does not live in the colon and rectum.
Doubtful. I doubt it.
No. Pink eye from viral conjunctivitis is highly contagious through physical contact. In your example, a person with it touches the infected eye, then touches your pillow, then you touch your pillow and touch your eye. Now you can get it!