I have been to every place on your list except Crystal Cave. Definitely delete Rushmore & Crystal Caves on your limited schedule. Wind & Jewel beat any privately owned cave hands down, imo. I liked Bear Country, but since you have no kids going, I think I would skip it also. Custer has much better wildlife.
You will need a day to go out to the Badlands. Tet gave you good advice about staying closer to RC, going out to the Badlands/Wall, and then making your base in Hill City/Custer. Keystone isn't a bad base either. A lot think it's touristy, but you will be so busy you won't spend too much time in town.
You will need another day to go out to Devil's Tower, and you can incorporate the Spearfish Scenic Byway either going out and coming back. You might enjoy Deadwood also.
You can spend a couple of days in Custer, but with your time frame, I would focus seeing as much of it as you can. As others have said, Sylvan Lake is lovely, the Needles Hwy is fantastic, and Iron Mountain Road has pigtail bridges (some even frame Mt Rushmore). The Wildlife Loop is neat and practially joins Wind Cave Natl Park.
As another said, if you go in summer, make reservations to see Jewel Cave. The tours DO sell out quickly in the summer, and you may arrive to find yourself with a wait of several hours. On your tight schdule, you don't want to waste time waiting around.
You will want to see Rushmore in the day and evening so that you can take in the Presidential Trail (day) and the lighting ceremony (evening). Crazy Horse has a fascinating history behind it, and I really enjoyed hearing the sculptor's story.
Have a wonderful trip!