if its just spots on objects were he sat or laid down it could be two things... a urinary infection (when this happens cats constantly lick their self which irritates the area and can cause sores / redness, just like a human who wipes constantly.) or it can be that he has a "war wound" of some kind from playing or fighting with the other cat, paw pads dry and cracking (both of which are treated with topical creams and antibiotics.

if there was something wrong inside him you would not find spots of blood around the house. it would be in his stool and urine, detectable through tests by the vet. example his urine would have the blood in it, or his feces would have the blood in it and in that case it would not be red spots, the urine would be tinged brown same with feces dark brown spots in it. when litter pan cats have this problem they usually dont go in the litter pan, they associate litter pans with the pain they feel so you would know it by having to clean up cat bathroom messes around the house.