I would definitely recommend that you go to see your primary care doctor
about this issue if you have not done so already. Although you are definitely right that bright red blood in the stool is often from a hemorrhoid, the fact that you recently had a large amount of blood come out should be evaluated. Especially in older adults (your question doesn't say how old you are) diverticulosis and other medical conditions can also produce bright red blood, and these can be more serious than hemorrhoids
The mucus might be a separate issue, but once again it is important to figure out where it is coming from. For example, the mucus could be mixed into the stool or it could be coming from the vagina or from the urine. This could give the appearance of mucus appearing with both urination and defecation, but probably the mucus is only coming from one of these locations.
Your primary care doctor will be able to perform a thorough physical examination and, based on what they find, help you decide whether or not any additional testing, such as radiology or blood work, is needed. Call your doctor at your earliest possible convenience!