You mentioned doing all this in the summer. If it were me, I would really rethink heading all the way down to the Keys in summer. Temps will be hot and incredibly humid.
And summer is definitely one time where you DO need reservations. Many people who say they never make reservations end up in Walmart and such places. I am guessing that you would do much better knowing you have a place to stay that has a pool or lake for the kids to swim in at the end of a long day. Also, your kids will want a place to run around in and a state or national park is a much better place to do that than a spur-of-the-moment camp place.
And while many people travel with pets, the poster is right that they will not enjoy the trip as much as you would without them. Also, you will not be able to leave them in your car or RV in summer, so this will limit your activities. Leaving them in the car in North Dakota for an hour or so is 100% different than leaving them in a vehicle for even 15 minutes in Florida heat in the summer. Even hikes can be difficult with pets because some places do not allow them on hiking trails, even on a leash. Ditto for most beaches. Consider what you will NOT be able to do with the dogs.
And ditto on all the paraphernalia, with a couple of exceptions. Do bring scooters and balls and such so the kids can run off steam at the end of a driving day! But do keep this stuff you put outside to the basics. You really do not need an outdoor run, and most times awnings are useless, depending on how you park. And use a checklist just before you pull out so you don't forget stuff.