CatB -- you may have already read these threads but, nonetheless, much information that both I and others posted within them addresses what you've asked as well as many more things you'll want to consider:
Grand Canyon: Rim to Rim interest? GRAND CANYON RIM TO RIM DAYHIKE -- OHMYGAWD! Snowfall closes the road seasonally to the North Rim every year, anytime from mid-October on. The ONLY hotel/motel on the North Rim is the Grand Canyon Lodge, which generally closes each year in mid-October. (There is another hotel/motel 11 miles north of the rim per se. Camping -- whole 'nother story (which I haven't researched). As stated previously in referenced thread(s) -- you may need to reserve up to a year in advance for the Grand Canyon Lodge. For some reason, I -- like smithb -- also lean to fall (late September/early October).
