As I read these forums, people seem to say AFFECT means only specific things. However, when my friends and I play, we just take affect to mean any card that would somehow AFFECT another card in any way.
Or, using the Dread Gazebo as an example, absolutely NO actions can be played on that base, meaning both ON BASE actions and ON MINION actions (hoping we have this one right)
For the first paragraph, it is very possible to play like this if you miss the first entry of page 8 of the rule book:
Affect: A card is affected if it is moved, returned, destroyed, or has an action attached.
As far as I have noticed, playing with a looser definition of affect creates a lot more rules questions with no clear resolution that the game term needed to be defined.
As for your second, unrelated paragraph due to the lack of affect anywhere on the card, Dread Gazebo does what it says, which is "Players can't play actions on this base." It doesn't say anything about minions on this base. (For an example of a card that does, see Hideout from the Tricksters faction.)