However, politics and social structure have never been my strong point, so I'm trying to figure out the broad strokes of what a low-tech society needs to hold together; What kind of offices are necessary in order to distribute the responsibilities of ruling, while also leaving the system relatively simple.
Well, you've got a wide range of possibilities but here're my thoughts.
Government in general consists of Soldiers and Judges. Historically there've been more of the former, but there's no real bar to an individual being both, especially at different times in their career. Each need to enjoy considerable prestige of some kind (it can be little more than fear, but without respect the society tends to be unstable). How this works depends almost entirely upon the nature of a society. Ancient Rome, with its powerful class structure/struggle between Plebians and Patricians, created an elaborate double judicial/legislative system to keep each other in check. They also had what was in effect private armies, at least until Augustus. Even afterwards, individual legions were often far more loyal to their commander than to the Emperor, and any sufficiently united alliance of said commanders could (and did) overthrow whoever wore the purple.
Europe in the Middle Ages built up a military aristocracy based upon heavily armed cavalry, the kind that required really serious capital to finance. In theory they owed fealty to a Lord, who in turn owed fealty to a higher Lord and so on to the monarch. But in practice there was very little sense of nation-hood, unlike (for example) Rome or to some extent Medieval Japan. The latter's military prestige was built around skill with the sword as well as mastery of associated skills such as composing poetry and flower arrangement (viewed as exercises in concentration, among other things). At the same time a kind of disdain was associated with thing like money, which was left in the hands of wives.
A lot depends upon the nature of the monarchy itself. Some nations (up to modern times) have a
Divine King, which usually limits the monarch's powers in ways both subtle and obvious. To be a religious figure means one must obey rituals, written or unwritten, or lose one's legitimacy and then people stop obeying you. The Mikado for example could not soil himself by having an income. Pharaohs needed to perform all kinds of religious ceremonies pretty much year-round. At one time the Chinese Emperors had to have sex with their wives and concubines according to a schedule drawn up by court astrologers. More subtly, a Divine King must remain to some extent above it all, lest they lose what some countries called the "Mandate of Heaven," i.e. the belief by others they are chosen by the gods. In Byzantium, it was simply assumed that any Emperor who was successfully deposed had clearly lost the Mandate--otherwise, how could they be toppled? Stalin and Mao both functioned as Divine Kings in modern times, surrounding themselves with an aura of being special--and in the process had to make sure there was an available scapegoat every single time something went wrong (given neither had a long-standing tradition upon which to build, they augmented this with a cult of personality as well as a campaign of terror against dissidents--Papa Doc in Haiti as well as Idi Amin in Uganda used similar means).
On a most basic level, governments of any complexity need a structure in place to maintain command of the military (including communications), some system of justice, a source of revenue, record-keeping of some kind, and representatives of executive authority, as well as some kind of legislative body (even if this is simply the monarch and tradition--but woe onto the monarch who dismisses tradition too much or too recklessly as Edward II, Aknaten, Nero, etc. found out the hard way--it can be done, as Peter the Great proves, but it is not an easy task).
In general a legislative body or council consists of those who have some kind of power base upon which to draw, which gives their word collective weight. Representatives of powerful Guilds, religious leaders, military commanders, wealthy and powerful families, etc. all are methods of getting such a seat. Individual seats in such might continue long after the original creator of same is dead, thus tradition favors such a group gaining power over time (as circumstances change, of course, said Council might turn into nothing but a powerless group that meets out of tradition more than anything else--not unlike the Senate under Trajan or the modern House of Lords in the UK).
