Does the Dread Curse of Azathoth spell stay active for multiple monsters?
From the base rules:


  Whenever an investigator enters combat with a monster, resolve the following steps, in order.


You enter combat separately with each monster. "Combat" refers to that set of steps (horror check, fight or flee, monster damage). So no, it doesn't stay active for multiple monsters, and yes, "this combat" means combat with a single monster.