No, they haven't. My team played in the Rose Bowl a few years back vs. the Pac-10 champs, Oregon State.
I have no idea how Illinois got in except by beating tOSU.
Yes, they do...with a couple of exceptions for the BCS system. If the exceptions don't come into play, the Bowl is an invitation by the Tournament of Roses Commitee for the Big-10 champion to come to Southern California and play the Pac-10 champion and for the Big-10 fans to come spend a week or two in the "Indiana Colony of California" and resort in the sun.