Are you collecting "taxes" and defending everyone in the commonwealth?
You're always free to make the declaration. The game provides no consequences for it if some faction disagrees with you.
Well your certainly in charge of the settlements, with or without MM involvement.....there is no option to take total control (DC and GN stay independent) however even when you win as one of the factions any future government is largely theoretical.
I assumed that they are actually trading for their food and that they are likely a location where raiders and gunners can trade since the population is apparently transient to a large degree, It would explain why the Raiders and Gunners at least never attack the place.
Sort of. At the end of The Institute questline, The Institute is basically the de facto rulers of The Commonwealth as they like to remind the citizens of Diamond City: ("The Institute is now the undisputed authority in The Commonwealth. Have a nice day.") And since my character is Director, that makes him, for all intents and purposes, leader of The Commonwealth.
So far I've just worked on building up the settlements. All of whom are flying Institute flags. But I consider them only really useful for resource extraction and research. The Institute is my only real concern.
Oh, and I wiped out Goodneighbor with my Synths.
As Lt Andronicus correctly observes, the Institute MQ leaves the Player as Director of the Institute and essentially ruler of the Commonwealth.
The BOS ending basically leaves Elder Maxson as the ruling authority, though he would logically make the Player his 'Imperial Governor', when he returns to the Citadel.
The MM ending, well, the player has about as much say as any of the United States original 'founding fathers' had. Which means there's probably a dozen other guys who will insist on getting their say as well. Constitutional Convention anyone? The Player has a good shot at becoming President when elections are held.
The RR, well, let's just say that after the euphoria of murdering the Institute wears off, about half the Commonwealth is going to want string you up for plunging the world into chaos and filling it with potential terminators. You can mitigate any negatives, as with all factions, by also being the MM General. However, it is only logical to assume that if the MM themselves do not finish the MQ, then their influence is much weaker on the Commonwealth.