PS: Whenever this happens (and let's be honest, we ALL did that at least once...) you can simply save & reload right there and then.
Rocks, boulders and other Korok Puzzle elements do not have a respawn cooldown - every time the zone is reloaded, they will be back at their starting position, and since you just saved, Link will be at the same spot too (instead of you loading an old autosave and then having to climb the mountain again or something like that...) This does not require a Blood Moon.
You can also just be sure to always carry a weapon or shield with a rock attached. You don't have to drop it or draw it, its position is counted even from your back.
That's hilarious. After the Korok cutscene, normally the rock seems to be removed (as a physics item), and a scenery "rock" replaces it in the circle. Does the rock stay on your weapon?
Yes. It stays on. There are videos of this on youtube.
However, if I recall it correctly, even the regular rock is not replaced and can be picked up after the cutscene. I don't know what happens when you revisit the site. And I haven't played the game in a while.
Immediatly after the cutscene, you can still pick up the rock. If you revisit the location later, the circle will be completed and the rocks can no longer be picked up, but the Korok will be still present. I guess it's some sort of safety measure so the player can't just repeatedly solve the same puzzle.