Reddit mentions of The Government of the Qin and Han Empires: 221 BCE - 220 CE

Firstly, sorry I cannot give you a detailed answer here. I did ancient Chinese as one half of my undergraduate degree, but haven't read anything recently (i.e. years).

However, I can direct you to some sources that I read which really helped inform me about the late Eastern Han/Three Kingdoms era.

• The Early Chinese Empires: Qin and Han by Mark Edward Lewis is incredibly insightful in regards to society, government, and military of the Qin and Han Dynasties, while his other book China between Empires: The Northern and Southern Dynasties deals with with post-Han China.
  
• Imperial Warlord: A Biography of Cao Cao covers the life of Cao Cao (obviously!) as well as the political situation he found himself in, which includes his position under Dong Zhuo.
  
• The Government of the Qin and Han Empires by Michael Loewe gives a fantastic insight into how the governments of the early empires was run!
  
  In regards to some of your questions, I'll take a shot at answering from memory (sources as above basically!)
  
  > if Dong Zhuo and others are so bent on being that powerful, why would they stop at Prime Minister?
  
  Dong Zhuo was thought to have been preparing to name himself Emperor under a new dynasty. But even besides that, we did try to rise higher than Prime Minister. He named himself Imperial Father, as in Father to the Son of Heaven (the Emperor Xian). This is important, as in Chinese political society where filial piety was important, the Emperor was the father of the empire, with only heaven as his superior. By naming himself the Imperial Father, he was de facto naming himself above the Emperor.
  
  In regards to regional warlords accepting the legitimacy of the Han Emperor while fighting each other, you need to understand where the political authority was seen to have originated, which was the The Mandate of Heaven (mostly). It was more politically difficult to get the rest of the empire to accept that you had gained the Mandate and the Han had lost it than to simply kidnap the emperor and issue decrees in his name (as Dong Zhuo and Cao Cao did). By acting 'under' Han imperial authority, warlords in control of the Emperor had more legitimacy to their actions than without him. This was made easier by the fact that the majority of later emperors in the Eastern Han were child emperors, who were the sovereign in name only, with court officials or eunuchs with real authority governing the state in the Emperor's name.
  
  Apologies I could not be more detailed, I have not read any of my books on this in a long time. But I think once my thesis is done, this question has knocked enough nostalgia into me to revisit them!