I've started three games as the Byzantines now. The first two didn't get beyond 1083 (the third one is only at 1088), but I saw some consistent opening moves. Resulting commentary:
Basileia ton Romaion
I usually delete Andronikos Komnenos from the Byzantine order of battle. This is partly because historically he died young, and partly because the game is constantly offering you new people to adopt into the royal family. Without Andronikos the first adoptee becomes the head of a junior branch of the royal family, but with him the first adoptee goes directly into the line of succession, with the result that the Komnenos dynasty will be very short-lived.
It would be nice if the Constantinople region had a land border with Nikaea province. With only one region connected to it by land Constantinople becomes a weak location for a trading center.
I would give the historical Romaion a "religious smith" personality over "sailor genghis." I agree sailor genghis is a good choice and probably the better strategy.
Constantinople starts with a fairground, which should allow three merchants (edit: or should it?). The Empire starts with only one merchant, but for some reason the game thinks they are at the cap of how many merchants they can build.
ai_label in descr_strat.txt is "default" but should probably be "catholics"?
I found that they have no idea of what to do with all their money. Basil II gave them rights over part of the Adriatic coast, so I think it would be reasonable to give them another province there.
In all three games their first move was to attack Cyprus with a nearly full stack of good units led by an elite leader. This is a poor move on many different levels: 1/4 of that force could easily take Cyprus while the rest takes the much richer target of Antioch, and that would not expose the pride of the Turkish army to being sunk when the fishing boat the AI puts them on gets attacked by a squadron of dromons. I don't know how to fix this, but it is a problem.
Historically the Turks had a long peace with the Byzantines around this time, as the Turks were busy assimilating their new conquests (and possibly with fights with other Muslim factions? I don't know), and the Byzantines were busy fighting off the Sicilian Normans. Giving the Turks a low reputation to represent their just having conquered most of Anatolia seems like a good representation of this to me.
You may want to consider giving one of the Muslim factions the "craftsman" personality, as that personality like to build races and health buildings, which are the faction-specific buildings of the Muslim factions.
Ballista_towers (data\export_descr_buildings.txt) take 3 turns to build. Everything else seems to take a multiple of 4 turns to build.
It would be nice if being at war with a faction of the other religion (i.e. Christian vs. Islamic) didn't hurt my reputation. It's not as if I have any choice - the Turkish AI wants war. And in general I'm not sure that Catholic nations' reputations should be hurt by being on Crusade, or that whatever unfortunate faction holds Jerusalem should find their nobles rebelling because they're at war with a billion crusading factions.
The HRE dissolved early in one of the three games, but was slowly prospering in the other two.
Council missions: the council_take_settlement mission (data\descr_missions.txt) has a turn_start of 20. This makes sense in the standard game where you spend your first 20 turns taking nearby rebel settlements before your borders start to meet your neighbours', but isn't necessary in this mod I think.