When two mastermind-controlled abilities have identical timings, the Mastermind determines the exact order of resolution, with Mandatory abilities occurring first
When two or more abilities with the same timing occur, any Mandatory abilities will occur first, followed by any optional abilities. In the event that two Mandatory or two Optional abilities have identical timings, the Mastermind chooses what order they resolve in. (This is clarified in Cosmic Evil's player rulebook, but the base game doesn't have this issue due to no mandatory abilities having the "Mastermind Ability" timing. Midnight Circle, despite having one mandatory ability with that timing, doesn't clarify it.)
Should an optional mastermind ability cause a mandatory ability's requirement to be fulfilled and the timing is correct for that mandatory ability, then the mandatory ability must immediately be resolved. In all cases where the entire timing of two or more abilities is identical and the abilities can (or must) occur, the mastermind chooses which ability occurs first and resolves it.
Abilities with a specific timing only occur a single time during that specific timing (i.e. The brain's ability can only occur once per day during the Mastermind Ability phase). Abilities with the "Always" timing are considered passive and, if a condition is attached, are in effect while the condition is satisfied.
In the example scenario Teofrostus provided, using the Brain's optional mastermind ability to add a second intrigue to the school causes the Factor's first mandatory ability's requirement to be satisfied. This means that from that moment forward, the Factor can do the same ability as the conspiracy theorist. Because the Factor can then do the same ability as the conspiracy theorist (an optional ability with timing "Mastermind ability"), the mastermind may then immediately place 1 paranoia on any character in the Factor's location (including the Factor).
Goodwill abilities occur in the order that the protagonist leader requests them.
In the example that Teofrostus provided, the Rich Man's Daughter's goodwill ability is invoked (and presumed to have taken effect) and a second goodwill token is added to the Girl Student. Because the Girl Student now has enough goodwill (2) for her ability to be activated, the protagonist leader may now use her goodwill ability to attempt to remove one paranoia from the Rich Man's Daughter (a student, and thus a valid target).
The rules very explicitly state that if multiple goodwill abilities can be activated, the protagonist leader gets to choose which ones he activates and in what order they are activated in. The goodwill abilities phase only ends if one of two things happen: either there are no remaining abilities that can be activated or the Protagonist Leader does not want to activate the remaining abilities.