Just to add to what others have said....Mt Rushmore is what you make of it. We just returned a few weeks ago from a trip to the Black Hills. Yes, there are four heads on a mountain! Major photo op, lots of folks taking pics and "selfies". But there's also a wonderful museum with great old photographs of the actual construction, the history of it, the politics of it (and if you think politics nowadays is bad...check out some of the political turmoil in getting this project completed!!) I thought that was fascinating. There's also a movie with narration about the whole project, and an expansion on some of the points of interest in the museum. There's also a hiking trail that one can hike, it's around the base of the memorial. There's also the sculptor's workshop that is interesting. And of course there's the obligatory gift shop for your shopping pleasure. Talk with some of the Rangers; they can give you some interesting perspectives as well, particularly the ones who've been there a while.
Regarding your doggie, when we were there, the outside temp was probably mid-80s, very comfortable. There's a multi-story parking garage, we parked on the lowest level and it was quite cool down there, and we were there mid-day. Other folks did the same thing, their fur family in their vehicles parked on the lowest level. If your pooch doesn't have separation anxiety, should do fine there.
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