The trip according to googlemaps is 2200 miles. We could (and have) do that in 4 days of getting up at 7 and being on the road until 8PM or so. Tiring yes, but our philosophy was to drive hard thru country that we weren't that interested in and then slowing down when you get to your target. The was before I retired, we take our time now. If you count the weekend on both ends of the two week block you actually have 16 days total.
As for St Louis, we will do our best to avoid the place. We drive right thru Atlanta and Houston but St Louis has horrible roads, potholed, patches on top of patches. And awful traffic. We were rear ended by a semi last time we drove thru.
I route us around St Louis by taking surface roads that go just to the east of the Mississippi up to Keokuk IA and cross the river there. As for the arch it's not worth the trouble when you are on a tight timetable.
Rushmore is similar, the approach roads are two lane winding roads and it will kill most of a day getting there and then getting back to the interstate. Trust me you will be surprised on how small Rushmore will look when you get there. Hold your hand out at arm's length palm towards your face. The palm of that hand would cover the carvings completely.
7 or 8 days in Yellowstone will just give you time to see most of the major areas. YNP is huge--3500 sq miles. Duval county is 900 sq miles for comparison. You will be doing a lot of driving and almost all of it will be in slow traffic. Every time someone spots a chipmunk they stop in the middle of the highway and block it for 20 minutes or so. Plus the park service is perpetually rebuilding the roads, Devote one day to the Tetons which is south of YNP and be sure to take the ferry across the lake there.