If you flush your eyes with water, will you experience sharp pain in the caruncle which would then result in bloodshot eyes?

Sure: You'd be better served using artificial tears or eyewash. The concentration (osmolarity) in tap water is such that it would probably be irritating to the eyes. Of course, if you get some chemical in there, tap water is better than nothing!

Not usually.: The water flush could dislodge or move a foreign body item which could be felt or painful (even an eyelash trapped). Otherwise, dry or sensitive corneas could be more symptomatic from the force of the flush or exposed nerves. Lastly, consider if the water wasn't "pure" or had any contamination or acid/base problems involved.