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Not mentioned yet, and it still doesn't really solve the numbers, but there is the matter of how much genuine control a hive world's governor has over the population. The stereotypical hive world's underhive is essentially urban anarchy. There is no attempt to gain control of this segment of the population, beyond sometimes skimming off some of the largest most successful gangs for recruitment into the PDF or Guard, and that serves more as a pressure release valve to prevent the underhive from boiling over and becoming too dangerous.

As imperial governance is usually depicted as resting within the hands of a few influential/ powerful nobles/traders I'd assume it is usually a cesspool of corruption. Take every stereotype throughout the centuries for decadent/corrupt governments in history. Distill them and add a healthy amount of 40k to it and you probably have your average ruling class.

Higher up imperial authorities do not care how you rule your sector, as long as you keep your charge loyal and subjugated and pay your tithes. With increased value for the Imperium itself you might face stricter controls. But I can very well imagine that planetary governors have no interest in getting rid of the underhive slums of their worlds. It might even be economically sound to recruit your guardsmen straight out of an underhive.

But I do like the point that handling the gangs of an underhive can be a delicate political maneuver. To either prevent a violent explosion on the (under-)streets, or even to maintain a balance of power for organised crime on these levels.

Just some notes on Haighus's massive guesstimation post:

 Haighus wrote:

1) Fuel:
Imperial Guard vehicles seem to run on anything, so their adaptability is good, but their efficiency is hard to predict. On the whole, it seems Imperial vehicles are much more efficient than modern ones, in part perhaps due to promethium. I would say fuel needs are no more than they are for the modern armoured division, and quite possibly less.

If there is one thing i would object to it is this. I came to a different conclusion. I put the standard fighting vehicles under the microscope from a realistic perspective in my project: (Most descriptive/investigative stuff is in the log on warseer).
If we use regular fuel as input, Imperial Guard vehicles would be only as good as WW2 ground vehicles. The power the engines produce for their given size is comparable to WW2 vehicles - which almost all used naturally aspirated engines. Example - Leman Russ Engine Phaeton pattern propelling a ~60ton vehicle 35kph. Engine room is in size comparable to a WW2 heavy tanks (e.g. Tigers). If the naming scheme of the engine is any indication there exist LR patterns with higher engine output (Mars Alpha Pattern Hulls) that use better engines - most likely charged versions. Supercharger (~1950/60 tech in ground AFV) or Turbocharger (~1970/1980 tech in ground AFV) can significantly increase power output for the same engine.
For fuel consumption consider that their standard AFV are really heavy. Leman Russ are heavy tanks for todays standard, and that's pretty much the only Battle Tank they use. Chimera and anything with that chassis is also around 40 tons, which is pretty heavy as well.
Empire of mankind is very fond of tracked vehicle designs in general. Almost all AFV are tracked. Many non-AFV are also tracked or track-like. So their percentage is higher in the motorized and logistical chain than in modern army.
Tracks increase fuel consumption since tracks have much higher resistance to movement and therefore higher fuel consumption than wheeled vehicles.

Just a nice example for the whole tracked vs wheeled logistical thing: consider that Soviets made giant Offroad Tank transporters (MAZ 535 and 537). It pulls a trailer (probably >10t) with their MBT on it (40-45t) on it, and it's own weight(22t). The Truck has the same engine as the tank it pulls (though less optimized), and they still considered this a valid strategy compared to the tank driving around on it's own.

 Haighus wrote:

Soldiers need more energy than people not fighting, so they have a bit more in terms of food than civilians (often around 4,000 calories per day). Note this is for the modern regiment- WWII regiments seemed to eat 35 tons per division, rather than 25. I suspect this is primarily due to improvements in packaging- much WWII food was canned, rather than vacuum packed in plastic, so that would increase the weight a lot. 35 tons vs 25 tons is a fairly huge increase though! I am going to use the lower, modern values.

The difference between modern and WW2 soldier is that there was a lot more physical work required back then due to lower percentage of motorized/mechanized forces and efficient equipment. Remember in WW1 german soldiers walked from german border to the frontlines in France at beginning of the war. Modern field systems are often self driven or at least have motorized tractors. In WW1 and WW2 a lot of stuff was simply carried, manhandled or pulled and carried by horses. Also equipment was much bulkier and much less weight optimized. No plastics, no advanced engineering for making everything only as heavy as it actually needs to be. Imagine pushing those 75mm and 88mm anti tank guns on steel carriage through russian mud all day, while wearing a backpack that has a steel skeleton that already weighs 6+kg on it's own. Imperial Gear is very much inspired by WW1 to Korean war (at least the old units - Steel legion, Cadians, valhallans, ... ). Just look at the vox caster. Soldiers in WW2 didnt have body armor back then. Imperial Guard do, esp. Cadians. So from physical demands i would expect an increase compared to modern stuff - because more personal carry weight than WW2 and about same material optimization degree.

As a sidenote (or key point depending on ground situation One thing unmentioned on a planet like this is drinking water... Very significant for logistics as well, because it has a big impact on infantry physical and mental performance, and they can't survive without it (as opposed to fuel, where units just become immobilized when it's lacking) It may be processed from local water wherever available - but to anywhere else it still needs to be transported around. And if the processing plants are disabled or supplies are cut off ... things look grim.

Another sidenote: Having to wear respirators, or worse, having to wear special protective clothing all day is extremely exhausting. The amount of sweating under even light physical activity in such sealed suits even under normal conditions is surreal. Now imagine full day full on activity in hot and dry conditions. So if it's really necessary to wear such clothing on armageddon most of the time then this will drive up food and water consumption considerably.

I found this (while looking for other stuff): descriptions of the logistics of the german armed forces in ww2 (including tonnage requirements and so on) - maybe it is of interest to anybody.
https://www.ibiblio.org/hyperwar/Germany/HB/HB-6.html

The main logistical advantage of the Leman Russ tank is its ability to run off of virtually any vaguely combustible fuel, meaning one isn't tied to things like oil fields or refineries. Efficient? Probably not, but it beats having an immobilized tank.

The Imperial Guard's most basic and standard equipment is meant to be able to function in virtually any environment off local resources so as to reduce logistical strain. That is why lasguns over autoguns or other solid munition based standard weapon for example.

The actual number of Imperial population may actually be much larger than the calculations (or possibly smaller). We don't know whether the Imperium actually has a million worlds, since like its population the Imperium has so many worlds nobody has ever bothered trying to count them. Given the troubles of its bureaucracy, it probably doesn't even have the capability to count the worlds it controls. Not to mention the fact that the Imperium likely loses or gains multiple worlds per day and so the number is ever-shifting.
The OP has a good estimate, but I estimate that the margin of error is in the several orders of magnitude. Not that a few orders of magnitude more or less really matters to a galaxy-spanning empire. The Imperium of Man is just so insanely huge it defies comprehension. I pity the Adepts of the Administratum whose task it is to try to make sense of it all.

This is a great thread.

I have nothing to contribute other than my sincere thanks for the read.