Others have much more experience than I, but I did recently travel from LaCrosse, WI through the Badlands to Yellowstone so I have a little advice.
Firstly, Sioux Falls may be a nice stopping place instead of Rochester: closer to the I90 and the falls are very nifty.
Instead of Pipestone consider Blue Mounds, MN; there you can see some prairie and a small bison herd ( although you will see many more bison later in your trip at the Badlands or Yellowstone!).
At the Badlands, you may be interested in camping at Sage Rim -- it is very near to where some bison like to graze. If your kids will be responsible around the wildlife it could be a thrill! It is a primitive campsite and you will need to pack in your water from the visitor center. The developed campground at Cedar Pass is nice as well.
The Black Hills, especially Custer State park ($20) are great but Mt Rushmore ($11) was only OK. There is really not that much to see in my opinion. We had a lot more time for this leg than you did so we enjoyed our stop (90 mins) but since you are pressed for time you may want to stay on track instead of getting bogged down here. Travel times were slow, especially on the extremely scenic SD 87 (recommended, but no RVs allowed, if you have a trailer then I might avoid as well).
Another thing you may want to save for another trip is Devils Tower -- we took a nice walk around it but it was pretty far out of the way!
Yellowstone: when we were there last week or so, a lot of roads were still closed, including the road into Grand Teton, so I can't offer you much advice there. Rt 14 through Shell Canyon to Cody is very scenic. Lots of signs say Rt 16 is the fastest to Yellowstone and they may be right, but the 14 is worth it in my opinion.
I will say that West Yellowstone, MT (west side of park) is closer to the figure 8 than Cody, Wy (east side of park). We stayed at both Old Faithful area and Mammoth and preferred Mammoth (more space, more "natural" setting). But the whole park was awesome!
One final word on camping -- it sounds like you are OK for equipment. We are in a basic tent and 32F sleeping bags and have no problem. You will want pads under your sleeping bags to insulate you from the ground, and a thick pad will also increase comfort. Otherwise, we primarily use a cooler, flashlight, camp stove/cookset (optional), and electric kettle. Also recommend a camping knife with a sheath so you can pack it away more easily.
Sounds like a great trip -- let us know how it goes!