I'd love to comment on these, as we were just there very recently. (See my trip report.)5-Sep Day in Badlands NP
6-Sep 2 hr drive to Wind Cave NP. Spend day.
7-Sep 1 hr drive to Mt Rushmore.
8-Sep Get early start! 3 hr drive to Devil's Tower. 3 hr drive to Bighorn Natl Forrest.
You can easily spend a full day at Badlands if you want to hike around. Hopefully they will have finished the construction at both the Visitor Center parking lot and on the main road. There are several good hikes from the viewpoints that will take you to further observation points. The bighorn sheep were mostly down on the western end of the park during the month of July, according to the rangers, and that's where we found them.
Badlands to Wind Cave is easily a two hour drive, if you are taking SR-36 and US-16A. Those are mighty slow going roads. We didn't realize when we used those to go into Custer after Badlands, that they would be so slow. Wind Cave below ground is a matter of taking the tours. We took the 9:40 am Fairgrounds Tour. Another in our group also opted to take the Wild Cave tour in the afternoon, which involves crawling around on your belly and means no cameras, phones, or other electronics. That wasn't for us. There are some places to hike on top of Wind Cave, too. You could also drive along Custer State Park's Wildlife Loop, too.
It's about an hour's drive to Mt Rushmore. We arrived early in the morning and really appreciated that, because there were less crowds. That shouldn't be as big a deal in September, but you never know! I'd recommend the Presidential Trail, which takes you underneath the monument. Depending on what you want to do, you may or may not need the full day. If not, go take the Iron Mountain Road or the Needles Highway drives through Custer State Park -- they are very worthwhile.
Your drive times are about right for Black Hills to Devil's Tower, depending on which route you take and where you were staying. We were in Custer City, went out through Deadwood, Lead and Spearfish, and it took us about 2-3/4 hours to get to Devil's Tower. If you take US-212, you may have the same -- especially if they haven't finished that construction! Three hours to Little Bighorn National Forest sounds about right, maybe more though. It really depends on whether you're talking about the Forest itself, or the National Battlefield.
What route are you using to get to Yellowstone? If going the Beartooth -- from Little Bighorn Natl Battlefield to Billings is an hour. Then it's another hour from Billings to Red Lodge. The Beartooth is going to be a 3-hour drive AT LEAST. We took 9-1/2 hours to go from Billings to West Yellowstone, MT via the Beartooth and stopping at the viewpoints in Yellowstone's northeast corner. (Unfortunately, it rained on us between Cooke City and the Lamar Valley, so we didn't stop between those.)