I don't have a cup but I don't see the problem in emptying your cup into someone's sink vs. shower... I'm also very low on the germaphobe scale. It's not like they're actively licking their sink or anything, right? (A kitchen sink would be a bit more touchy.) But seriously, I'm sure there's been other gross stuff in that sink before like blood from cuts, vomit, hair, snot... they're designed to get rid of liquid waste.

And if it's my personal bathroom, well, my sink is pretty old/stained/slightly gross as it is so someone's blood in it for a few minutes isn't going to make it any worse. :P