I think it sounds fun, and I wish I would have done something like that when my kids were younger. On your long driving days be sure to schedule plenty of breaks. I drove for 8.5 hours for the first time in July. I only stopped for gas, and I was exhausted by the time I reached my destination. In the future, I will not go more than 3 or 4 hours without a break of at least an hour to stretch my legs and rest my arms.
I drive a 24 foot Class C, and because of my husband's work schedule, I often arrive at campgrounds several hours before he does. If I am dry camping, I can be set up within 10 minutes of pulling into my spot depending on the site's levelness.
Cooking while driving might be tough. It might be easier to prep food in advance, and you can probably even break that up by prepping stuff during stays with friends and family. I like to precook things like shredded chicken and ground beef, seal it up in food saver bags, and then freeze it. Put in the fridge to thaw before you go to bed, and the next night it only takes a few minutes to make tacos, spaghetti, chicken salad, etc.
Consider teaching the dog to lay on a mat when he's outside with you. That will keep him from carrying in a bunch of dirt that was trapped in his fur.
Lay out what you want to take and then try to cut some of those items from the list. I'm always amazed at the end of the season how much stuff I didn't use. Don't forget cards and a couple of board games. You might also consider a notebook to use as a travel journal.
Make sure you know how to use all of the RV systems,a d it might be a good idea to be in places with a good internet signal for the first few nights just in case you have to google something that you can't figure out how to work.