We did this trip just about a year ago, and went from Duluth so a similar route as you. We took three weeks in total.
Suggestions: Definitely go via the U.P.
We went out through North Dakota and came back via South Dakota, just to not travel the same route. In North Dakota you can visit Theodore Roosevelt National Park and Medora. Medora is a great place. TRNP has a couple of campgrounds that are pretty basic, but OK for an overnight or a day or two.
Definetly recommend visiting the Badlands whichever route you go, but a drive through there may be sufficient. Also, you may want to visit the nearby Ronald Reagan Missile Silo museum (a history lesson of the cold war for the kids).
Definitely see Mt. Rushmore. Again, a history lesson for the kids. You can see Rushmore and Crazy Horse together in a day. But give extra time (a couple of days) for Custer State Park. Lots to see and do there. The Wildlife loops, and be sure to take the kids for a Chuck Wagon hayride/sing-a-long/bison dinner. A great experience.
A visit to Devil's Tower is worth it, but you won't need to revisit there on your way back. The campground at Devil's Tower (in the park, not the KOA) is a lovely little campround if you need to stay there.
There is MUCH to see and do in Cody and you will want at least a couple days there. The Cody Museums are again a great history lesson for you and the kids.
We used Cody for a simple overnight stay on our way to Yellowstone. We actually stayed overnight for free at the Cody Walmart (there were about 30 campers there!). Then we stayed at the Cody KOA for a couple of days on our return route and spent time at the museums and in town. Be sure to visit the Harold Schafer museum in Cody-you won't believe how much stuff this guy invented!
The times that you have at Yellowstone are probably adequate - 1 week to see most of the main attractions.
And I agree with those who say to leave the dog at home. In the National Parks, there are strict rules and very limited places where a dog can go, and you will be gone all day. Either kennel the dog at home, or get a relative to care for it. This trip is about you and the kids. The dog will not really appreciate seeing mountains and for sure won't appreciate seeing the inside of your 5er for 16 hours a day.
Have a great trip. Whatever you ultimately plan as your itinerary - it will be a trip you will long and fondly remember.
