Unless you're making a pasta salad, you should NEVER rinse it in cold water.
Fresh horseradish is so much better than the kind in a jar. I'd suggest your use a food processor; grating it by hand will make your eyes water incredibly. It's very, very hot; much hotter than the jarred kind. Don't buy gigantic pieces of root, and don't grate much at a time; it loses its heat pretty quickly.