For example, a notorious cheat/glitch is that you can use the "save and quit" feature to essentially re-roll Markets, Relics, etc. Would using the save and quit feature to do so violate the game run? It's within the mechanics of the game, technically speaking.

It's also worth noting that this may not even be relevant to speedrunning, as the time taken by saving and quitting could outweigh any time saving benefit by attaining a better unit

I don't know if I'd call that a glitch but I don't see why that wouldn't be ok. I don't know if it would be a time save though at least in the Joan of Arc run. The last mission can save about 30 seconds if you get the relic in the top right corner of the map so I don't know if the save/quit method will save that much time. I say try it out though on other missions. Why not, that's the fun of speedrunning.

I had the thought to start mission 6 (Joan of Arc) first since its the most crucial with RNG and wait till you get the relic in the right spot but the rules for the run are: you start the first mission and end on the last one so that wouldn't work.

Yeah I mean I was reluctant to mention this since I haven't seen anyone else employ it, but one colossal benefit of saving and quitting besides RNG is actually... Castles.

Using save and quit, you can train 4 units on a castle per day (by saving and quitting 3 times).

Castles work out to be around 3/4 the price of 4 buildings, plus Castles enable you to build the typically OP civ-unique unit, plus Castles can be built near the enemy base, and that castles take 1 villager 1 turn to build, while 4 buildings obviously takes 1 villager 4 turns, which could be spent instead building farms, Mills, mines.

Using this tactic, I've been able to complete notoriously long and difficult missions like Retaking Jerusalem in 16 days (Mameluke and Longswordman spamming), and Yashima (samurai spam) in 18 days. I'm pretty sure the advantage of being able to swarm the enemy early on with civ-unique units saves more time than saving and quitting itself.

Though this benefit only really applies to missions where you either start at Age 3, or you need to reach Age 3 to have any chance of winning (Cyprus, Yashima, Retaking Jerusalem, Supreme Ruler, Empire Expansion, etc)

Not even mentioning how this benefit scales with the n of castles (2 castles = 8 samurais produced in 1 turn compared to 2 samurai)

Well I totally misunderstood what you were asking in the first post then. The save/quit glitch is not something I have ever heard of but I just tested it out in terms to castles and it totally works. I don't know how you figured it out but it was awesome to test out. That should improve those longer missions for sure. Having 4 strong troops in one turn essentially should speed up the speedrun.

I believe these runs are Any% so glitches like this would be allowed. Maybe if the moderators see this post they can say yes or no to confirm.

Just adding my own vid I made on this for context/visualising the glitch: