The mirror being up likely refers to the viewfinder mirror. I haven't used a DSLR, only mirrorless, but I wonder if that could be independent of the shutter.
That's exactly the mirror that gives (D)SLR's the R ("reflex") in their name. It's in the path between sensor and lens and redirects the image towards the view finder. When taking a picture this mirror moves out of the way and the image is projected on the film/sensor instead. The advantage of that system is/was that the photographer can preview the image as it will get captured.
Correct, it’s like a mirrored fan blade and rotates continuously.
Ironically, with mirror reflex system what you see through the finder are precisely the moments not captured on film. Very quick events, muzzle flashes for example, can be missed entirely.
Notably, some Bolex cameras (and others?) used a beam splitter system where you do see what the film sees, with no flicker from the shutter. The tradeoff is a dimmer image in the finder, and you’d need to overexpose to compensate for the lost light.