Here's my short list of must sees for the area:
Custer State Park
I added Spearfish Canyon since you are not looking to see any caves. I normally list Wind & Jewe Caves on my must sees, but it's fine if you don't want to see them. Caves aren't for everyone. Spearfish Canyon and Devil's Tower make a great daytrip.
The most centralized places to stay are Keystone, Hill City or Custer. Rapid City is a bit too urban for my taste, and staying there will add about 30 minutes drive to most attractions.
Here's a link to a relatively recent thread about Bed and Breakfasts in the area:
This will give you some ideas on what is out there. I have not stayed in a B and B in the area, as we do hotels.
4-5 days will be ample time to see the sights of the area. September is a nice time to visit, as crowds are down, and lodging prices typically decrease as well.
Dining is available in the small towns. Rapid City has local eateries plus some normal chains. I personally like Blue Bell Lodge in Custer State Park, Alpine Inn in Hill City, and Big Time Pizza in Keystone. The other lodges in Custer State Park are highly rated as well.
To get a good feel for the area's attractions, visit travelsd.com and order a vacation planner. There's great information there, and you will get a much better idea on what to see and do. The area has so much more than Mt. Rushmore.