Yes, the 30 day time limit applies to both FQ and diversity cases. But, the one year bookend only applies to diversity. So if a case based solely upon state law hangs out in state court for 13 months and then the complaint is amended to add a federal cause of action, the defendant could still remove provided that the case satisfies the requirements for federal question and supplemental jurisdiction, but the 30 day clock would start running from the date of amendment. But, it is hard to imagine a situation in which a state court complaint could be amended so late to add a drastically new, jurisdiction creating claim. This would be pretty prejudicial, since discovery would have been proceeding under state rules and now federal rules might apply.
You are correct that the one-year bar in diversity cases makes it possible for plaintiffs to keep a diversity destroying defendant in the case for a year and then drop them just to make sure they stay in state court. Some courts have said this is BS and should not be allowed. (See 327 F3d 423 and the accompanying law review pieces found when shepardizing up). I think this would make a great "what do you think" exam question.