3- Most likely Flagstaff. It's roughly an hour as I remember from the canyon with lot's of hotel space.
2- Take the Oak Creek Canyon route from Sedona to Flagstaff, its a great ride, scenic and you get to see Sedona on the way.
1- I'd take a leisurely trip up the Oak Creek Canyon route, stay in Flagstaff overnight and then see the canyon the next day for as long as you like. Just read your post again and saw you have one day, that is a tougher one, stop at the visitors center, grab a map and do your own tour or sign up for one but you may have to sign up far ahead to get into a guided tour, good luck.
4- We did not avail ourselves of any tours, etc. but drove around and look down a lot. It's amazing but after a while it seemed to be another big hole over and over, personal feelings as many are devoted to it. We felt Monument Valley AZ was more fun and quite amazing on its own, plus you see it in backgrounds from John Wayne westerns to Forrest Gump (where he stops jogging, its in the back ground, can't miss it once you experience it). It even shows up in animated cartoons.
Have a great time. Bonding at special places is a wonderful thing to do with your children.