Question 1 for high school and first-year university:
I have once again revised my proof for this problem. I assume that each complex number must be distinct, otherwise, one could arrange all of the numbers in a line of length 2, which would not form a rectangle yet sum to zero.
By rearranging ##A + B + C + D = 0##, one may derive that ##A + B = -C + -D##. This means that for each two complex numbers, their sum must be mirrored by the sum of the other two complex numbers. This is depicted in the diagram below, where the cyan line must mirror the yellow line and the pink line must mirror the green line.
For any two unit-length complex numbers that are distinct and are not opposites of each other, the origin, their sum, and the two numbers will form a parallelogram where each side has a length of 1. This means that the line drawn between the two complex numbers will be perpendicular to the line drawn between the origin and their sum. Therefore, the sum of the two unit-length complex numbers will be an angular bisector of the two.
Imagine trying to rotate one sum independently. The sum must remain the angular bisector of the two numbers that it is the sum of. In order to do this, one must rotate both of these numbers by the same amount that one rotates the sum. In doing this, one will cause the two other sums perpendicular to the sum (in the case of ##A+B##, these would be ##A+D## and ##B+C##) to no longer be angular bisectors. Any rotation of two or three sums may be broken into moving each sum individually and therefore will not work either.
Decreasing the magnitude of one sum independently increases the angle between the two numbers that it is the sum of. Analogously, increasing the magnitude of one sum independently decreases the angle between the two numbers that it is the sum of. This is impossible to do without causing 2 other sums to no longer be angular bisectors. Scaling three sums together by the same amount may be interpreted as scaling two sums and then scaling one independently (which does not work as previously explained).
This limits one to only rotating all of the sums together or scaling the magnitudes of two opposite sums by the same amount. Rotating all four sums together would rotate all four numbers together, which would not affect whether or not the numbers form a rectangle. For the four complex numbers to form a rectangle, the angle between any two complex numbers must be equal to the angle between the other two. For example, the angle between ##A## and ##B## must be equal to the angle between ##C## and ##D##. By altering the magnitude of two opposite sides together, one simply increases/decreases the angles within each of the two pairs of numbers by the same amount. From the arrangement in the diagram above, any transformation will preserve the equivalence between the angle of any two complex numbers and the other two. Because this condition will always be satisfied, the four complex numbers must form a rectangle.