Some random thoughts.
LIBRARY RULES.
Triamore is supposed to have a very good library. It is one of the main themes of the covenant. However, it is so because the emperors of the Holy Roman Empire send it their books. These are mostly books they get as gifts. So we will have a lot of MUNDANE books, but not necessarily supernatural books. Falconry books? Take a shovel and dig away. Lives of saints? Yup. Magic Lore? I think we have one of those somewhere in that back cabinet. An initiation script? WTF is that???
For rules, I think we will use the rules in Durenmar (GotF) for abilities. Or just a lot of build points.
Expertise areas: falconry, theology, area lore, single weapon, social graces (nobility), management. Civil and canon law. Maybe some Legend lore...
The library will also bring studious people to our doorstep and act as a secondary source of income.
Regarding diversity of initiation scripts and then like, that is one of the tough parts of a multiplicity of traditions. This is one of the reasons that make the hedgie order less advanced than hermetics. And why some traditions try to create a comprehensive theory of magic. 
THE CODE OF HERMES.
The swearing of the code is part of the initiation script of Parma Magica. The script predates the current membership of the Order, but due to mystical reasons everybody that undergoes the initiation understands what they are swearing. So, if you have Parma you have sworn the Code and take it seriously.
ARBITRATORS
There are a bunch of magi that perform the role of Quaesitores and Arbitrators. They do not intervene unless someone asks for them to intervene in a conflict or uncovers evidence of wrongdoing on the supernatural side and alerts them. So they are less nosy than the official quaesitores. One Arbitrator is always present in a Tribunal gathering (when at least 3 covens gather to settle disputes), and he will act as mediator and head judge.
TRIBUNALS.
Tribunals are way less formal than in the official setting, and generally much more local in scope; most of them cover a pair of duchies at most. Their frontiers are way less formal and can overlap. Basically they are places to get deals and solve disputes you want an Arbitrator (or a bunch of crazy hedgies) to mediate in. That means that you might need to attend several tribunals if you have issues with people indifferent directions. I make this so we can crank up or down hermetic politics as much as we want.
THE LEGACY OF THE ORDER OF HERMES.
There are rests of the OoH around. The books you know about that talk about magic from the OoH are total gibberish. Even books written in monasteries that talk about that order read like a madman's rantings after around the year 1000, and before that year if they try to talk about their magical abilities. Most (educated) superatural traditions know that there was an Order of Hermes and that they invented the Parma. It seems that they were powerful and created a system of tribunals not very unlike the one you have today. However, there seems to have been some curse or something like that, and no item or book that he arcane writing in it seems to have endured. Few books remain, and those are either in a very advanced code (all of them) or they have been corrupted somehow.
Some of the books were written as far as late 10th century, but they are undecipherable, and after a while it seems that the Order of Hermes ceased to be. Some current nobility have their origin in this mythical order, but nobody pays them much attention, like the kings that use Arthur or Sigfried as their foundational mythos. The counts of Fengheld, descended from really powerful magicians? Yes sure mylord, whatever you say.
How about that? You surely guess what flaw the curse caused. Only in overdrive mode compared to then official rules.