Sky King has good suggestions for bikes. Cross, Mountain or any bike with a fatter tire/more tread is good.
I will underscore lights. The two mile long tunnel gets very, very, dark and very, very cold. Do not think you can ride in your partner's light beam. It doesn't work very well. Headlamps or a bar mounted light are good. I've ridden it several times if you need other thoughts.
FWIW, we never use the shuttle. We always go down and back.
From what I've head the rental bikes aren't that great, more like comfort bikes that mountain bikes.

Some people combine a visit to the Hiawatha Trail with the Trail of the Coeur d'Alenes.