Let's break down your travel by day:
Day 1 – leave Connecticut; drive 6.5 hours to Niagara Falls
Day 2 –leave Niagara Falls and drive 9.25 hrs to Evanston
Day 3 – Evanston
Day 4 – Chicago
Day 5 – Chicago
Day 6 - Chicago
Day 7 – leave Evanston, drive 1.5 hrs to Milwaukee, drive 1.25 hrs to Madison, drive 1 hr to Wisconsin Dells
Day 8 – leave Wisconsin Dells, drive 3.5 hrs to Minneapolis, stay
Day 9 - Minneapolis
Day 10 - Minneapolis
Day 11 – leave Minneapolis, drive 9.25 hours to Watford City (Badlands)
Day 12 – Badlands
Day 13 – leave Badlands, drive 5.25 hrs to Mount Rushmore, drive 7.25 hrs to Grand County
Day 14 – Grand County
Day 15 – Grand County
Day 16 – Grand County
Day 17 – leave Grand County, drive 2 hrs to Denver, airport, drive 45 minutes to Boulder
Day 18 – leave Boulder, drive 6 hrs to Arches National Park
Day 19 – Arches
Day 20 – leave Arches, drive 5.5 hrs to Zion
Day 21 – leave Zion, drive 4.75 hrs to Grand Canyon
Day 22 – Grand Canyon
Day 23 - Grand Canyon
Day 24 – leave Grand Canyon
For the most part, it's doable. My biggest concern would be Day 13 – if you leave the Badlands at 8 a.m., you’ll arrive at Mount Rushmore about 1:15 p.m., then you have another 7.25 hours to drive before stopping for the night. If you spend 2 hours at Mount Rushmore, you won’t get to Grand County until after 10 p.m.
I'd skip a day in Grand County and make an overnight around Mount Rushmore.
Going with a 26ft rig or smaller is a good idea. The kids will adjust to sharing the bunk - they're only going to be sleeping together, so as long as one doesn't kick in his sleep, they won't really notice the other one.
Getting on the go can be very simplistic, especially since you plan on using the RV's kitchen (not setting up a grill, etc. for every meal. Keep your campsite minimal - a chair for each person - and use the campsite picnic table.
When you prepare the RV for the trip, make sure everything has a proper home. You don't want to have to unload a bunch of gear just so the kids can get into the bunk. Use packing cubes for clothing - each person gets a color. That makes it easy to get their stuff without confusion in the mornings - grab your appropriate color cube and head to the bathroom. Use the campground facilities for dressing - 4 people dressing at the same time can get awkward. If you can't find a place to properly store an item, think twice about taking it along. The more you have to move around each day, the harder to keep things organized. Get everyone on board with putting things away as soon as they are done with it - no letting dirty dishes sit to soak, no putting something on the table because "I'm going to play with it again". If it's not being used, it gets put away.
Cooking on the road. Some folks will fix meals in a slow cooker while they're driving all day. You'll need to figure out your power needs - battery bank, generator, solar, etc., if you expect to use electrical (slow cooker, microwave, etc.) appliances regularly. You mention having the kids cook while you're driving - walking around in an RV while on the road is awkward. Fixing a sandwich while sitting at the dinette is one thing, standing up at the stove cooking a hot meal is completely different. A slow cooker that will fit in the sink (so it can't fall off the counter) is an easy solution. Preparing meals ahead of time (make a couple large batches of easy to re-heat meals on the days you're not traveling) and using the microwave to heat them up will greatly simplify your meal times.
Definitely do a few trips before the big one. Try a long weekend trip and shoot for a place 300 miles away, stay two nights, then drive another 100 miles away, stay a night, then drive home. That will give you an idea of how you will really handle the driving - be prepared to make adjustments to the big trip after this. Do a 4-7 trip to work out how to arrange storage, morning routines, meal times, etc. Those two trips alone will do a ton to prepare you for a month in the RV.