There appear to be many free hotel shuttles between McCarran International Airport and the Strip or the downtown. If I don't book a hotel, can I still use the shuttles? Can I also get back from some hotel on the Strip (or in the downtown) to the airport?
No, these shuttles are provided for hotel guests, not for anyone to use. Also, some cursory research by Googling seems to show that most major hotels on the strip do charge their guests for using the shuttle. The hotels that provide free shuttles tend to be not the ones you think of when you want to stay on the strip.
Those that do offer a free shuttle service seem to all include a disclaimer similar to this one (happens to be the first one I found on Google, emphasis mine):
Silver Sevens offers complimentary shuttle service to and from the McCarran Airport, exclusively for registered hotel guests.