"Cut into halves" vs. "cut in half"

The bulk of your question seems to be quoting an answer to an interview question about cutting a rectangular cake into two equal pieces when a rectangular piece has already been taken. For the first bolded item, “It’s easy to cut a rectangle into halves”, one could just as well say “It’s easy to cut a rectangle in half”; and similarly for “This will cut both rectangles in half”, one can say “This will cut both rectangles in halves”. In the particular cases in this example, either form can be used; by habit I'd probably use half.

The term halves refers to the parts that result when something is cut in half. In the interview problem, the important thing to emphasize is that each rectangle is cut in half, or sliced into two equal-sized parts. You can emphasize that via either wording.