However, how would I mod my own AI only through a Starcraft map?
You wouldn't. There's a way to load a modified aiscript.bin using EUDs, but I don't remember the process, and I'm never making content for SCR again (even if I did, I'd still be modding). If you choose not to engage in modding you won't be able to use the extended limits blizz offers (since EUDs prohibit them) and your tools will be severely limited, so all I can say then is good luck.
Does this mean that the script requires only one random AI-owned building or AI-owned beacon to work? And with that minimum threshold, the AI will attack and move units?
Area town scripts take ownership of buildings (and overlords) that are present in the location they're run at. They stop executing commands once all buildings (and overlords) they own die. As long as at least one building (or overlord) belongs to the town, it will remain active, and keep executing commands.
Could I run Terran, Zerg, and Protoss AI's in the right order and get the AI to attack with a tri-racial army?
Should be fine but do some testing, depending on the scripts you may run into issues. Anything with start_town at the beginning of the script shouldn't be run alongside another script with start_town. Notably I've seen zerg campaign [difficulty] scripts stack without leading to crashes but they tend to lead to performance issues for the AI, so I always advise against it, especially when the tools to write your own (much better than blizzard's) AI exist.
I think making a custom AI is currently beyond my level of expertise
Nah, if you understand EUDs to any degree, you can learn aiscript super easily. The most annoying part is figuring out why a certain thing hangs the AI up (especially if you're editing vanilla AI instead of using the aiscript extender plugin via modding) but that becomes easy enough through trial and error, and you have the luxury of being able to ask people here and on various discord servers for assistance.
After all, I have one computer AI that I ultimately want to competently fight up to seven different players across multiple continents simultaneously.
I experimented with co-op vs AI maps in the past and I generally find them to be pretty fun, though obviously in your case you're dealing with a diplomacy map where the AI is going to be using mechanics the players won't have access to (i.e. unit production). To that end I have no idea how feasible it would be to make stock AI perform adequately under those conditions, and I once again encourage you to mod instead of trying to hack a few half-assed blizz scripts into something that resembles a challenge.
