Mountain driving is definitely different than flatland motoring but no worse than driving in the city of Chicago, IMO. In fact, I would rather drive in the mountains and I am from the Chicagoland area. My husband does not care for heights and has no problem driving in the Black Hills. Speed limits are low and if followed, there will be no problem. No way will you be able to completely avoid narrow and/or winding roads in the Hills, but it is possible to take roads that are less winding. However, you will miss out on some fantastic drives, two of which are on the top ten list of most scenic drives in the US - Iron Mountain Road and the Needles Highway.
The route to Mt. Rushmore from Rapid City is an easy drive. There are several ways to continue on to Custer State Park from Mt. Rushmore. The least winding way would be to go back through Keystone and continue on US16 to US16/385 to Hill City and continue south to the town of Custer and then east on US16A to Custer State Park. A more scenic route, and shorter, would be to take Hwy. 244 west from Mt. Rushmore and hook up with US16/385. This route is somewhat winding but not too bad. 16A through parts of CSP is also somewhat winding but not as hilly. The Wild Loop Road is very easy to drive. To get back to Rapid City from Custer State Park you can pick up Hwy. 36 going east to Hwy. 79. 79 is totally flat all the way to Rapid City.
I highly, highly encourage you to put your fears aside and drive both Iron Mountain Road and the Needles Highway. I don't think you will regret it. The scenery is absolutely incredible. There are many places you can pull aside for photo opps or to let traffic by if someone is making you nervous. The best way to experience Iron Mountain Road is to drive it from the south to the north. Stop at the Peter Norbeck Overlook for a breathtaking view of Mt. Rushmore and the surrounding vista. Keep in mind that traffic is usually very light, compared to metropolitan driving, and that touring buses and large motorhomes navigate these roads all the time. You will be in a rented sedan or SUV. A piece of cake!