Nose bleed

It is possible that your nasal passages are dry, thus making the capillaries that are near the surface easy to crack resulting in the nose bleeds. Do you live in a dryer climate or does this tend to happen more during one part of the year? You should use saline drops, but not all of the time. They can actually cause more dryness because your body thinks it does not need to produce any mositure in that area. Running a cold air humidifier at night will help provide moisture and give time for healing.

As far as getting bloody noses with running water - I'm not sure. I tend to think that it is an illusory correlation meaning that there really is no relationship between the two, but when you notice you have a nose bleed, you look for running water (or you are in the bathroom and may notice it in the mirror) and that the correlation between the two may not exist.

Here is a little info about illusory correlations to see if maybe that is what is going on:
http://en.wikipedia.org/wiki/Illusory_correlation