Why is my cat's nose swollen and bleeding? He's shivering and sick.

Sorry to hear that Monroe isn't acting like himself. Based on the symptoms you have described and his age, I am inclined to think that something infectious may be the cause of his current predicament. Upper respiratory infections can definitely result in all the clinical signs you have reported and are extremely common in cats of all ages. Other causes of true bleeding from the nose may involve something traumatic, primary nasal diseases, tooth root abscesses, and certain cancers of the mouth and oronasal cavity. If you are able to obtain a temperature on Monroe at home, I would suggest it (look for between 99.5-102.5). You can also take a look at his gums for a moist and pink appearance. Dryness and paleness may be indicators of blood loss or dehydration. Regardless of your findings, I would certainly recommend having Monroe looked at sooner than later particularly if the bleeding you are seeing is significant in quantity. Hope this helps!