It seems there is the Emperor > Prime Minister > Provincial Governors as a basic outline of the government. I know the book is fictionalized, but take Dong Zhuo for example.
As a provincial governor, he deposes the current emperor and places someone he's more keen on in his stead. He gets the consent of the other officials to do this through their fear of him. Dong Zhuo then places himself as Prime Minister (which seems to be the position that wields the real power). Then you have Cao Cao trying to kill Dong Zhuo and restore the original line of emperors (and he eventually becomes Prime Minister I think because he's as brutally ambitious as Dong Zhuo...I haven't gotten that far yet but remember from some videogames based on this time period).
But meanwhile, it seems like the governors are always fighting each other for power even though they all mutually accept the authority of the one emperor. They raise huge armies to slaughter one another at times. So it seems like they are allies in the sense that they bow to the same emperor, but are at each other's necks at the same time. How could this system work?! (Feudal Japan was like this as well, at least in one period (correct me if I'm wrong). All the lesser leaders were all trying to fight each other while simultaneously pledging allegiance to the emperor.)
Could someone clear all this up for me?
And also, if Dong Zhuo and others are so bent on being that powerful, why would they stop at Prime Minister? Why not kill the emperor and establish their own dynasty? Would this be too bold of a move, basically ensuring a huge rebellion against their own ambition?
edit: I've read some more since posting this (so much happens in this book!). A confederacy led by Yuan Shao is created to oppose Dong Zhuo's tyranny. Many high officials and main characters lead armies in the confederacy. Sun Jian leads the first battle, but is denied supplies by Yuan Shao because of fear of Sun Jian's ambition being like Dong Zhuo! Sun Jian's army becomes decimated as a result. This is the kind of contradictory nature of the political system I'm talking about, where even in a confederacy aligned to destroy an oppressor, they are all constantly throwing daggers in each other's backs.
Then Sun Jian leaves the confederacy, followed by Cao Cao leaving too after his own massive defeat. Cao Cao goes back to camp and realizes everyone is just planning their own schemes for domination (including himself!) and the confederacy crumbles. I'm now at a chapter when Yuan Shao has turned his army on another leader's army of the confederation, Gongsun Zan.
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.
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!
I will try to explain the governing system of Han dynasty at the time of three Kingdom era, although I'm not too well-read in this matter.
At the highest level, the Imperial Court a.k.a the Cabinet of Han China is headed by the Emperor. Prime Minister is a position in Qin - early Han period; sometimes there may two of them - called left PM who takes care of civilization affair and right PM on military affair; sometimes, just one. The PM(s) is the highest officer in the Court, followed by other Ministers. Occasionally, there was no PM but a position called 相国 - General of the Nation, which is pretty much the same as PM but more important (just sound more important since PM is already the highest position you can possibly be - other than being Emperor). It is not uncommon to see the PM has more power than the Emperor himself. However, PM position is latter replaced by "three dukes" in charge of administration (司徒), military (司馬) and supervision (司空) respectively. Yuan Shao often brags that his family has been keeping at least one of these 3 posts for 4 generations straight.
When Dong Zhou seizes power, he is appointed to be the duke of supervision, then switch to the duke of military and general of the front - in modern language: "The secretary of defense and Chief of Staff". And then he became the General of the Nation, which is a position that no long exists but Dong just wants to make his point that he is not just a de facto, he is officially the most important guy. When Cao Cao controls the emperor, he also starts as the duke of supervision; later, he reinstalled the position of PM (for himself of course) and modify the 3 dukes to smaller positions to consolidate power for himself alone.
Then at the local level, Han China is divided into provinces (州) -> counties (郡) -> districts (县). Earlier in Han rules, each province is headed by a governor (刺史 or 牧, depend on the period). In theory, they are just public servants who manage daily administrative works. They have to report to the Imperial Court and only have limited access to military power. Governors do not have the autonomy to, like, raise huge armies to slaughter one another at times; not when the Imperial Court have a strong army of its own and able to call every other governors to purge the traitor.
At the end of Han dynasty, the political system is decentralized. Governors are given more power; meanwhile the imperial court is in constant power struggle and loses its firm grip over military authority. The Yellow Turban Rebel forces the Imperial court to rely on armies raised by local governors, therefore, empowering the governors even more. Then Dong Zhuo after a series of events, find himself to be the one staying in the capital city, with an army - therefore he can control the Emperor and the Imperial court. Now all the local power see how that goes, they are not happy (for reasons which may less related to loyalty than they said); the campaign against Dong Zhuo gives local governor the excuse to raise even bigger armies and to ignore orders from the Imperial Court. At this point, it is pretty much clear that every men for himself; since the court can't really do anything to keep them in check. Governors have became warlords.
Edit: grammar and some incorrect number.