You can set it with eye shadow, this may cause sensitivity to the eye though. Otherwise, just try different eyeliners, different formulas may be longer lasting on the waterline than others. I prefer Mac fluidline or NYX eye shadow pencils.
Yes, if it is waterproof. If it is a pencil make sure to sharpen it before and after every use in there as it shaves of the layer that touched your eye. If you use a brush be sure to wash after...
The best way to do this is taking a Q-tip dipped in eye makeup remover. Gently move it back and fourth over the eyeliner. This isn't scary as it sounds, LOL. try not to apply long-lasting eyeliner on...
If you don't wear contacts, wearing eyeliner in your waterline is safe. (If you do, there's too much chance of scratching or knocking them out of your eyes to do this.)