MMR can shed through stool. A mother, who has a son that received the MMR vaccine reacted poorly (she got an acute form of it which caused her to get MS). Her doctor thinks it's from the virus that shed into his stool and she was still changing his diapers during that time.
Yes, and it can shed via the urine as well. Secondary transmission of measles has never been documented. Doesn't mean it hasn't happened as with the above posters story, but it is probably fairly unlikely.