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Yup, the search space has been completely explored. You can read more about the effort to do so here: http://www.cube20.org

This MIT open courseware lecture explains more about the optimum move sequence to solve a Rubik's cube. Some people call this perfect sequence of moves the "God path." https://youtu.be/s-CYnVz-uh4?t=25s

Yes. We know the greatest lower bound distance to any solution (20 turns) and we can always find the minimum length solution.

Edit: maximum -> GLB

Rather than "maximum", don't you mean the largest minimum is 20 turns? I'm pretty sure I can do 21 turns and not solve it.

Yea, it's the maximum over the minimal solutions for each configuration.

Yes, edited, meant greatest lower bound.