* The proof uses the Axiom of Choice, so no, it's not constructive;
* The boundary of the pieces has to be unmeasurable, in that sense it's similar to Vitali's set[0];
* No, the pieces are more-or-less a "fog" of points.
* I don't know why you have "proofs" in quotation marks, and I don't know what you mean by the second half of the sentence (even if I replace "contraction" with "contradiction").
I do feel like a careful reading, or perhaps re-reading, would let you ask more specific questions that we can help you with.
> I don't know why you have "proofs" in quotation marks
Because proof by contradiction is viewed by an entire category of mathematicians as "wrong" (please make sure to note the use of quotes in case you missed them).
"Proof" by contradiction, tertium non datur, proof by the absurd, whatever you want to call it, more often than not entirely fails to produce working examples, which makes the "proof" fare less useful than the one that explains how to build an actual exemplar.
* The boundary of the pieces has to be unmeasurable, in that sense it's similar to Vitali's set[0];
* No, the pieces are more-or-less a "fog" of points.
* I don't know why you have "proofs" in quotation marks, and I don't know what you mean by the second half of the sentence (even if I replace "contraction" with "contradiction").
I do feel like a careful reading, or perhaps re-reading, would let you ask more specific questions that we can help you with.
[0] https://en.wikipedia.org/wiki/Vitali_set