Grew up using and around about a dozen dry sump outhouses. Some exposure as an adult to wet sump outhouses.
Lets build a list of problems and what to do to avoid them.
1. Sides of the hole caving in, being eroded in or being dug in by pets and wildlife. Always great fun when you have the family of skunks living in the hole under the outhouse. Would build a bigger base and a bigger roof as first step. Maybe add some buried chain link to keep animals from digging in.
2. Water in hole. One location was a high water table and another was next to a ditch that flood about once a year. Both stunk to high heaven and had major fly problems.
3. No place to wash hands. Had to go back to the house to wash hands
4. Cold/hot and in a few cases snow drifts inside thru cracks in the wood work.
5. Stinky and bugs at the best of times
6. Other?
The best one we ever had was what my mother called a WPA outhouse. Floor and toilet riser were poured concrete as a single pour.(factory made?) Riser was funny shaped long skinny hexagon. Seat at one end and a riser vent pipe (was about 6 inch stove pipe) that was behind your back sitting on the toilet at the other The concrete was probably 7 or 8 feet square. Walls standard 2x4 framing with ship lap style siding on the outside and 2 small windows in the front wall up high. Had some big square cans with a round paint can style lid mounted on one wall for TP storage that kept it dry and mouse and bug free. (Other best answer seen was a bunch of Shortening size cans that held one roll each on a set of shelves. The cans were nearly a snug fit on the shelves so the mouse couldn't get to the plastic lid.) In a low use outhouse this would be high on my list of design features. The same big square cans also held the lime for odor control and a scoop. Never did use anything that used carbon while growing up. It was all lime here. One final feature of this concrete slab was both ends were curved up and it had built in tow loops so the concrete base was also the skid. Since this was intended for the tow the outhouse 6 or 8 feet over a new hole in the ground which is no longer legal anywhere this part doesn't matter. Now to do this one better I would concentrate on making it so it was easy to hose out. Concrete was a bit hard to clean so I would paint the riser and floor and possibly lower walls in a really good grade of epoxy paint in a really light color for best lighting. This riser was sort of in the middle. If it was moved to the side a bit very little more area would have been needed to bring this up to wheel chair accessible.
Was in a outhouse in south western Wyoming that had a tight sealing metal door that came up from the bottom about 6 inches or a foot down. The toilet lid up top was connected to it and when you opened the lid up top to the over center vertical position it moved the lower door clear out of the way so nothing could get to the door to stick to the door. Close the lid up top and the bottom door was spring loaded to close and seal over the bottom. Looked commercial design so might go looking for it.
First lets talk heating and cooling options first.
1. I live in a passive solar home so I am totally for the standard single slope clerestory design with long over hang to the south for heat control. Be aware this roof design needs real over hangs at both ends to work properly as well as the south. On a house this is easy but for an outhouse this will 3 or 4 foot "porch" ends out both sides and 3 or 4 foot overhang to the south for your sun control eve. The problem here is the building needs to be high thermal mass inside so concrete, rock or rammed earth with lots of insulation outside. There are permies compatible options here.
2. Second option would be to convert the south wall to a modified version of a trombe wall using a window screen collector, VIG max glass and a folded path collector design. Build the vent doors into this so summer operation you just thermal siphon the air thru it keeping the air outside the building. Change 2 quick vent doors to change from summer operation to winter operation. 2 lessons learned from my current collector. Vents need flicker, rodent and bug screening as well as a replaceable filters to keep dust out. You will still need a way to open the glass occasionally and clean it. (say ever 2 to 10 years)
3. Third option, What about heat pipes down into the ground. Where you intend to go back in a hill they would be even more effective. They are completely passive with no moving parts. Some systems come with a 50 year waranty and life expectancy up into the 100 year range. Would be pricey to start but if your building was well insulated could probably do all the heating. Would heat to within 5 or 10 degrees of your soil temperature and are completely passive. Basically a one way heat diode moving soil heat up. The online information shows an individual 3/4" heat pipe capable of up to 120 watts. Modern full size homes well insulated and designed the talk about being able to heat with 1500 watts for a full home. So an outhouse with 1/10 or less the size with a number of heat fanned out for more ground area would probably heat fine the rest of your life. Build some into and insulated sink cabinet also and you could probably keep pipes from freezing even if the door was left open.
4. Earth tube. the Ceres' greenhouse information stated that 83 feet of 4" drain tile pipe buried 8 feet would bring in air continuous all winter long and never reach freezing at the indoor outlet of such air. Longer runs yet would come closer to earth temperature. Where I am that is 56 degrees. So the building could probably be kept at 45 or 50 degrees with nothing more than power for a small fan.
Now that brings us to odor control, fresh air intakes etc.
Most of the wet sump outhouse use a large diameter steel pipe, mounted on the south side of the building painted dark black as a chimney for odor control from the biological side. Some put a spinning turbine at the top that runs a fan so it sucks any time the wind blows. The turbine also keeps birds out too. Now from other problems I would possibly add 4 functions to the base of it. Air filter, bug and rodent screen, an antiback draft door for wierd wind conditions and possibly finally a charcoal odor filter. Now I was around one of that the part down next to the building has a mirrored slightly concave area to increase the heat that reached the lower pipe.
Now the inside system needs to pressure up. For winter daytime operation this would be easy to do with some solar thermal air. Or with a PV powered fan on an earth tube.