Few things:
- If you're looking to drive cross country, IMO, only places worth doing that are mountain range areas. Either Rockies in West which is scenic all over through many states, or Appalachians in East with places like the Blue Ridge Parkway... everything else is boring including that drive from Florida to Dallas which is mostly on I-10. Nothing but swamps, worn down towns, and hurricane damaged forests with maybe the highlight driving by the USS Alabama in Mobile and that's about it.
- I second the people who suggest visiting Nashville to get your Country Music fix. Spend a weekend. Stay near Broadway Street which is famous for its music bars. Visit Country Music Hall of Fame, Johnny Cash Museum, and more... Hardest part here though will be your 10-year-old. She can come with you during day, and Broadway Street is still going on, but if you want to see all the live bands at night and the festivities, definitely not for kids. Treat yourself out to some authentic Southern or Soul Food. Memphis Fried Chicken!
- Theme Parks, you'll need AT LEAST 3 days in Disney, 2 days in Universal to get a somewhat fulfilling experience, and I would recommend 4 for Disney, 3 for Universal if never been before. If going in summer, very hot, add a day or two for recovery. Daughter will probably like Magic Kingdom at Disney the best, and the Harry Potter Worlds at Universal, but tons of stuff to do in both parks. Get your daughter the Disney Princess outfits, and treat her out. She's only a kid once. I would skip the rest of the tourist parks unless your daughter is a lego fanatic, she might enjoy LegoLand. Seaworld only worth it if you go to their specialty parks where you can swim with dolphins and do other cool things that generally would take an ocean/beach trip to experience.
- Space Center. Easy one hour rental car drive from Orlando to the Space Coast. Think it's worth seeing at least once. If you want to experience some cool Beach Life stuff, that is not the place to go. Go down South towards Fort Lauderdale or Miami, but probably only worth it if was only you and the miss.
- New Orleans. Always worth a visit. But again.. not very kid friendly. Things to do in the day, but like Nashville, adults only in evening. Mardi Gras time right now if you're going soon, and always worth attending, but book a hotel NOW. FYI, a lot of crime and drugs in that area. Treat yourself out to a great Cajun Food place. Of all the cities on this list, this one is by far the most unique.
- Dallas.. for sure the Cowboys and JFK experiences are two things worth doing and city is known for. Generic metroplex, but will have everything a large metro should.... good museums, a lot of rodeo and country music stuff since it's down South, good sports teams, big amusement parks, etc... Fort Worth probably has better rodeo, country music tourist stuff if that's what your wife is looking for. The entire DFW metro is huge so make sure you look at attractions in all surrounding cities. You're in Tex-Mex territory so look for some great Mexican Food joints.
- Arizona, now you're starting to get to an area worth driving. You're where the Rockies start to begin. I'd probably fly out to Phoenix or a big city in the state and get a rental and plan it out from there. All types of sites. Grand Canyon is the big one, and that is a large area, so make sure you go the main Southern Station. That's the real Grand Canyon. Monument Valley which extends into Utah, you'll recognize if you saw any old school westerns. The famous Meteor Crater is in Arizona. Hoover Dam in Las Vegas along with Red Rock Canyon and of course.... VEGAS baby! Montezuma Castle, Vermilion Cliffs, etc.... Just search a list, and pick what you want to see and how it will work out logistically with the driving. Sedona is a great looking city both natural and manmade, and cool place to shop. Of all the places on this agenda, this is where you'll get the most impressive photographs.
Long write-up, sorry, but love to travel, and been to all these places more than once. So only way to make it work if you want these places and only have 3 weeks would be...
9 days Orlando, Central Florida.. flying
2 days Nashville.. flying (not on your original list but you specifically want to see country music culture, and no better place)
2 days New Orleans.. flying
2 days Dallas.. flying
6 days Arizona.. driving
I'd probably cut off Nashville, New Orleans or Dallas and spend more time in the 2 of 3 you don't cut. Hope this helps.