Is it possible to have a hemorrhoid that bleeds but isn't painful at all?

Yes. Internal hemorrhoids can bleed without any pain at all. Internal hemorrhoids are located above the "dentate line." this is where the lining of your colon meets the skin of your butt (similar to the difference between the outside of you lips and the inside of your mouth). Above the dentate line there is very minimal sensation, so irritated internal hemorrhoid tissue can bleed without pain.
Yes, internal H's. Internal hemorrhoids are the ones that bleed; external ones can swell and if thrombosed (clotted) can cause pain. However, rectal bleeding should always be evaluated, so see your doctor for a proper diagnosis.