The trick to having the dishwasher work is to run the tap on the sink until the water is hot. Using powder or liquid dish detergent instead of those sub-optimal expensive tablets also helps, as does leaving sauces on some of the dishes or cookware (only scraping off solid chunks of food)
Even cheap dishwashers clean very well (assuming no clogged filters or mechanical faults) if you follow the above steps.
Dishwashers may not be as fast as going by hand, but the idea is that you free up time requiring active attention by using the appliance. Dishwashers also use much less water for a cycle than 99% of hand washing setups.
Probably just fairly bog standard geometry combined with some material science/engineering on the physical properties of the cardboard (such as how much it can compress/stretch)
Geometry: this problem more or less boils down to a 2d analysis since we need merely to look at a circular cross section. You could calculate the area each roll takes up by calculating the outer circle area minus the hole's area, then divide the hole area by the ring's area to get a theoretical maximum. This is assuming the material cannot stretch or compress. Not sure if this has a name, but it probably does.
Material science: Maybe you could measure the dimensions of a roll, stress test it in various ways, and re-measure the new dimensions to get a profile of how the cardboard warps. You could use that to get a better estimation of how much cardboard you can stuff into itself, but I'm not as sure on the details there.