I spent October 03 - October 04 at Oxford, getting a M.Sc. in Medical Anthropology. I had four Rhodes scholars (three US, one Canadian) in my degree program, which only had 15 students in it total.
From talking with them, I gather the experience of applying and going to the state/regional interviews is very time consuming, especially if you're applying for other scholarships. A number of them applied synchronously to medical school though, so I don't think timing is a problem if you're organized and on top of your work load.
OP, you definitely have the numbers for the Rhodes, and your athletic ability will also help, as others have said. The committees definitely prize community work (volunteering, organizing, political work, etc.) as well. If youre interested in the Marshall, which has a different aim than the Rhodes, its helpful to have published as an undergraduate.
Feel free to PM me if you have questions about the process or about Oxford. I didn't go through it myself, but I have about ten friends (some got it, some didn't) who (I'm sure) would be willing to give you some advice on the process.
Good luck!
