I am always amazed that you professional don't get tired to answer us amateurs.
Because usually it's short quick questions

First (37 part) Q:
You can't compare a cement board installation to a mud bed installation as the mud bed is a heavy reinforced structure resting on the floor of the home. Cement board panels are too thin and too light and are not monolithic across the floor and therefor need to be bonded to the structure. They do allow for a minimal amount of movement around each screw, enough not to be bothered in a home. The thinset underneath of the cement board is not a bonding layer, it's a bedding layer and that is why most cement boards allow either modified OR unmodified even though unmodified without an additive shear pretty quickly from the subfloor or underlayment beneath it. Because it's only a bedding layer is also why you can install cement board over osb subfloors even though all thinsets say can't use the thinset to bond tile to osb. This small amount of movement in the cbu panels is why the joints need to be taped, otherwise you can get cracking every 3'x5'.
The fleece backing of the Ditra is bonded to the floor with the thinset but between the lower level of fleece that is bonded to the floor and the upper level of the fleece that is bonded to the plastic of the Ditra, the fleece in between is able to allow the tile and the floor below to exist independent of each other, sort of like engineering an old mud job into a 1/8" thick roll.
Second Q.
It depends how you look at price. Based upon just square footage, Ditra is much more than cement board. Ditra at HD is about $85 for 54 sf. Say you have 200 sf to install. You need 4 rolls, that's $340 and 2 bags of thinset to put it down at $15/bag, that's a total of $370. It would be 2 trips to carry it all in to a job site. Installation time would be not much longer than the time it takes to comb out your mortar as any cuts are quickly and dustlessly done with a utility knife. Edges are butted together and small scraps can be used to fill in wherever needed.
Cement board
15 sheets at 40lbs and $9/sheet, $135
Cement board screws, 800 pack $25
3 bags thinset $15/bag $45
150 ft roll cbu tape $6
Total $211
You saved $160. Now you are going take 17 trips to get it in, hit a wall, ding, now you got a repair to make, Hope it's not up too many steps, you are already tired from loadng the cart at the store, unloading the cart into your truck, and getting the cbu at least near where you will be installing it. Comb your mortar, lay the board, start driving in the screws, all 58 per sheet. How many times does the screw not want to countersink and you are backing it out, blowing away the dust brought up by the hi/lo threads and resetting the screw?
What you save in materials cost with cement board over Ditra you spend several times over with time. That's considering things at regualar retail, most of us are getting things at better pricing than that.
So, Is Ditra less costly? Depends on how you do your accounting. Is it easier? Significantly. Less work? Significantly. More solid? Since neither are structural, it's not a good way to phrase it. Better engineered with better performance characteristics? Yes.
More solid only comes with more plywood and Ditra with 3/8" plywood underlayment will arrive at a lesser thickness overall than 1/2" cement board and still offer a lower transitioning height.
Third Q) Out of time, I'll finish later, but all 1/4" plys always must go. They are specifically excluded from ceramic and stone installations.