>It would be interesting to understand why load increases more quickly than thrust for increasing blade counts.

The key to this is that, for both ducted fans and props, a larger swept area is more efficient, while # of blades simply changes the torque/rpm ratio with negligible effect on efficiency. Thus, for a given torque, it's always better to drive two longer blades, than 3 shorter ones.

If you enjoy looking at pictures of WW2 fighters, you'll notice that the early planes had 2 bladed props, the midwar ones had 3 bladed props, and right at the end they jump to 4 and 5 blades - the reason being that the better engines got, the more torque they had to dissipate, and you can only make a propeller so big before you hit ground clearance problems. Helicopters, on the other hand, can make their rotors as large as they like - and so they do, and typically only have 2 blades. Only on helicopters like the Chinook, where they made the blades as long as they could feasibly engineer and still had torque to spare, do you see 3.

> Do ducted fans have similar changes in load-to-thrust ratio given an increase in blade count?

Not really. The parameter that describes how many blades are in a ducted fan is known as "solidity", and while it does have minor implications for blade shape the general efficiency is excellent no matter what. It's actually surprisingly insensitive - you can take a ducted fan designed with a 5 bladed rotor and just slap a 7 bladed rotor in there instead, without even redesigning the blades, and provided the power source is equally efficient at a somewhat lower RPM you basically won't be able to measure any difference in performance at all. Ducted fans can turn shaft power into air momentum with about 90% efficiency.

The reason that large swept areas are more efficient than small ones is simple physics and geometry. Energy (the thing you put in) is 1/2 mv^2, while momentum (the thing you get out) is simply mv. So for best efficiency you want to keep v as low as possible. It's better to get your momentum by moving a lot of air slowly, than a little air quickly.