How can a 3x5 rectangle be cut (and still remain in one piece) and be able to be folded into a 1x1x1 cube?

How about a 3x3 square?

I'm not sure what KS intends, the second case is already covered in the original

Two Cube Folding Puzzles.

And the first case, all 11 distinct ways of unfolding a cube into a hexomino fit in a 3x5 rectangle, and 10 of those hexominos will fit in a 3x4 rectangle, too. So it is rather trivial to make redundant cuts and fold the excess onto the hexomino before folding it into a cube.

I did see opportunity for something not covered by the original problem: How to cut and fold the 3x5 rectangle to doubly cover the faces of a unit cube. That is to say each of the six faces has at least two layers of the original rectangle covering it.

I found this answer. The solid lines indicate where the cuts should occur and the numbers indicate which face of a standard die is covered when folded.

+--+--+--+--+--+

| 6| 2 3| 3 1|

+ +--+ + + +

| 2| 2 6 5| 5|

+ + + +--+ +

| 1 4| 4 5| 6|

+--+--+--+--+--+