Using the Playful/Trickster spell defensively means that most of the damage in Fizz's kit is negated. This means he, as an assassin, has to chose between his safety net and getting the kill on you. ( Plus it has a very high CD and high mana cost in early ranks, which means he can be outdueled if you measure and time it.) It can be flashed, as can his Q and R moves. He has a lot of counter play. DoT spells do indeed continue on him when he's in the i-frame, like ignite or malzahar aids or other burns which make him still killable in iframe so long as its on him before he goes in.
Fizz has a very easy to predict play style pattern as well. If you're having trouble with him, the best thing to do is play a few games on the champion so you understand these limitations and use them to your advantage. It's funny that you mentioned Fizz's passive as OP since its considered to be one of the most lackluster ones in the game, which is why I think that might mean you just need to spend a little more time with the fish to feel what's built in there to balance him.