Part of the Soviet advantage in tank design over the Germans was that in Germany, the Wehrmacht had very strong influence over all stages of production. That could produce a tank well suited to the actual needs of tankers, but it also meant a lot of tinkering, a lot of line stoppages, a lot of incompatible parts. I've heard the German tank industry referred to as "artisinal" before.
It was less that German military oversight created the less than mass manufacturing industrial base and more that the German economy was set up differently and was operating on what the civilian structure had created in the post-WW1 ToV restricted heavy industries. For one thing Germany is a quite small nation with exceptionally good infrastructure, so having huge factories where mass manufacture could happen as in the USSR or USA at the time was sort of unnecessary, as small subcontractors could easily get their subcomponents out to larger facilities very quickly, while in large countries like the US or USSR transport was over large distances, so you needed huge facilities in one place that did everything (or nearly everything) under one roof. Plus the German economy was designed to rapidly change it's production to service international export trends and simply could not compete against the US with it's huge mass production model. Furthermore the restrictions of the ToV meant that armaments manufacturing was still being rebuilt when WW2 started (they figured they'd be done rebuilding the industry in 1942 assuming there was no slow down in rearmament spending that was starting to hit in 1939), while the tariff wars of the 1920s-30s meant that the German economy was relatively stunted compared to where they were at relative to the other European countries and the US in 1914-18. All that coupled with army demands for quick improvements in models on the production lines at the drop of a hat meant that production was limited compared to other countries like the USSR or US, but generally they were producing much more reliable, longer lasting models than say the USSR (google the US army's report on the T-34 in 1942). The Soviets even turned captured German tanks into modified weapons for their own use due to how much more reliable they were than Soviet tanks in the first half of the war:
http://www.tanks-encyclopedia.com/ww2/soviet/soviet-su-76i.php
The Soviets, OTOH, had far less army micromanagement in production. Tanks would be built to a certain spec and mass produced on those lines. They could make greater use of efficiencies, gain line experience, and get more tanks to the front faster. So even though on paper the Germans should have been able to outdo the Soviets in tank production, their system couldn't even best the Soviet system in efficiency.
Not exactly the case. Their biggest advantage was simply having huge facilities where production was done under one roof and were given the order that output was more important than reliability due to how huge losses were; why build a harder to make, but long lasting tank when it's survival was measured in weeks? So due to being able to make effectively knock off versions of their own tank the Soviets could make a lot more than they even should have been able to due to being able to cut corners to ensure output. As the war went on and things stabilized Soviet production quality improved due to a variety of things including experience, lack of needed to cut quality at the expense of output, L-L aid (think machinery and even raw materials), and phasing in upgrades.
Also even on paper the Germans weren't set up to produce en masse like the Soviets were, so I'm not sure where you're getting that from. Hitler and various German officials marveled at the huge Soviet tank facilities they overran in 1941 that dwarfed anything they had. The secret recording of Mannerheim and Hitler from 1942 specifically mentions that; Hitler even apparently told Guderian in 1941 that if he had known intel on Soviet production was correct he might not have invaded.
The non-Soviet examples are rather limited though. Most WarPac command economies were restricted to copying Soviet designs by politics, since the Soviets didn't really want the WarPac states to develop independent armaments industries for... obvious reasons. The one exception was Czechoslovakia, which had loose enough restrictions to go with some of it's own designs... and also did a pretty outstanding job of it. Moving outside WarPac, North Korea, Vietnam, and Cuba did respectably enough given their countries size, resource constraints, and external issues. The Chinese example is a bit strange as Mao's buy into the People's War concept led to a relative de-emphasis on mechanized warfare that required such an extensive armaments-industrial base and the post-Mao leadership was more interested in moving away from command economy to build the economy with a view of reinvesting the gains from that economic reform into the military a few decades down the line (as they are doing now).
With that said, I could see how the success of Soviet command economy when it comes to their military-industrial complex probably does contain a degree of historical-cultural context. The idea that it was an aligning of the traditional strengths of Russian armaments industry with the quirks of a command economy that allowed the Soviet armaments industry to be relatively innovative and productive (in glaring contrast to the rest of it's economy) is a thesis I can get behind. One could apply it to the Czechoslovak example as well. That the Chinese lacked the same sort of industrial-armaments background in their history at the time the command economy was dominant is likewise a point in it's favor.
Basically yes. None of the communist states outside of those with a history of quality weapons production were able to take advantage of the command economy mass production model. Mao's China also had the serious problem of lacking an industrial base capable of producing mechanized warfare weapons/equipment, so only developed it from scratch and Soviet knockoffs. Going forward perhaps they will develop their own innovations, but thus far it doesn't seem like they've really broken from existing design paradigms.
I know that in cars, a transverse engine gives you more interior space for a given exterior size, so I guess it's a good thing for tanks too?
That was basically what the Soviets found with the T-44 vs. the T-34. Also they apparently first experimented with that transverse engine for the T-18 tank in 1928, which was an upgraded Soviet version of the French FT-17. Though only a small tankette, they were able to cut weight and get a better tank out of it, even if it was just an upgraded WW1 model. It gave them experience to use on making their own designs, which they then were also able to parlay into applying the transverse concept into the T-44.