11 out of 12 random positions are unreachable.
This first part just looks at the positions of cublets, without regard to their orientation.
It is possible to place the corners in all $8!$ position permutations. Separately, it is possible to place all edge pieces in all $12!$ position permutations. But it is not possible to apply a permutation that swaps only two corners or two edges.
This is easily seen. All possible permutations arise from the application of 6 basic permutations corresponding to quarter-turns on each face. Each of these quarter-turns cycles 4 corners (an odd permutation, equivalent to 3 swaps, an odd number) and 4 edges (also an odd permutation). The combination of these two cycles is an even permutation (3+3=6 swaps, an even number), so any series of basic moves also results in an even permutation. This sets an upper bound of $\dfrac{8!12!}{2}$ position permutations.
It is possible to construct permutations that solely cycle the positions of 3 corners or 3 edges, so using these 3-cycles you can independendly reach all $\dfrac{8!}2$ even corner permutations and $\dfrac{12!}2$ even edge permutations, resulting in $\dfrac{8!12!}{4}$ total even position permutations. Turning one face throws both the corners and edges into odd permutations, and from here using the 3-cycle moves you can reach all $\dfrac{8!12!}{4}$ odd position permutations. Add these together and you get a lower bound of $\dfrac{8!12!}{2}$ reachable position permutations, which meets the upper bound.
In the second part here I'm talking about changing the orientations of the cubes, without changing their positions.
Corner rotations:
Moves exist that jointly counter-rotate any two corner pieces without affecting any other changes. This sets a lower bound of $3^7$ possible rotations of the corner pieces.
Consider a rotation labeling system for the corner cubes. Mark one facelet of each corner as special. Each corner cubelet position has a face whose normal points upward or downward, and call this position 0, the one clockwise from it 1, and the next clockwise 2. You will find that each basic move preserves the sum of these orientation numbers modulo 3, which sets an upper bound of $3^7$ possible corner orientations.
Edge rotations:
Move sequences exist that jointly flip any two edge pieces without affecting any other changes. This sets a lower bound of $2^{11}$ possible rotations of the edge pieces.
If you put together an arbitrary orientation numbering system like the corner orientation numbering system above, you'll find that all basic moves preserve the sum of orientations modulo 2. This sets an upper bound of $2^{11}$ possible rotations of the edge pieces.
So the cube in total has $\dfrac{8!12!2^{11}3^{7}}{2}$ permutations reachable through basic moves.
If you disassemble a cube you can place the corners in any of $8!$ position permutations and $3^8$ orientations, and the edges in any of $12!$ position permutations and $2^{12}$ orientations. This totals to $8!12!2^{12}3^{8}$ possible configurations, which means 11 out of 12 potential configurations are not reachable through basic moves.
Best Answer
(This answer only discusses under the context of the MIT paper OP referred to.)
I have done a research on "why the heck did they put commutator and conjugation in Rubik's Cube notes" (written in a more polite language of course) and read tons of materials related to Rubik's Cube.
Basically, there is no theoretical significance now. It is useful for practical reason and stays there for historic and educational reasons.
For people who has attempted solving the Cube, I'm sure it happens at some point of time that you say to yourself "let's put this cubie away using move $A$ so the next move $B$ (which usually does something helpful) does not affect it, and we can bring it back later with a reverse move $A^{-1}$". This is conjugation $ABA^{-1}$. Commutator has similar practical meaning.
They are useful when people want to figure out by themselves how to solve the Cube because they are simple and the effect is obvious to eyes, so they are used extensively in last century when people do not have the computational power to look at the mathematical essence of the Cube. I believe Singmaster's Notes on Rubik's Magic Cube and possibly one issue of Hofstadter's Metamagical Themas that introduces Rubik's Cube have significant influence on this.
Right now the academia is not really concerned about how to solve the Cube anymore since you can easily find some algorithms online. Rubik's Cube is mostly used for demonstrating or introducing concepts in group theory, and given that historically people often see commutator and conjugation in books about Rubik's Cube, it is natural to pick them as two of the concepts that are demonstrated by Rubik's Cube.