How to play vs. assassins after lane phase:
1. Wards.
2. Positioning.
That's literally it. Maybe build QSS or GA if they're fed and you're a carry. If you've got poke in your comp, focus on the assassin. If you're squishy and high-value, don't go near the front or places where you don't have vision. If you're a tank/bruiser/tanky support, stay the fuck where you're supposed to be: in between your squishies and the enemy carries.
It also helps to play assassins enough to understand the mindset and recognize situations where they can go in, so that you can react as soon as they do. Know how champions work -- in the case of Fizz especially, make sure that whoever on your team is most tanky will be in position to eat his ult away from everyone else. Don't let your squishies get close enough for him to Q them; force him to burn his E to get into range. As soon as he loses his invinc frames, CC him and blow him up (just like any other assassin).
The only reason an assassin will ever be able to pick someone off from a grouped enemy team before a fight starts is if they don't know how to position. Even half-decent positioning will force them to wait for someone else to engage before they go in, which leads to another way to counter assassins: don't let their team engage on you. If you've got an engage comp, go on them -- either they'll blow their burst on your frontline, or it'll be delayed while they try to reach your backline. But given UXLZ's statements about Ezreal's E, I'm not certain whether he'll actually get the point about how important positioning is.
If you've really got trouble dealing with Playful/Trickster (or Vlad's trollpool), here's what I suggest (and what I did): Play the champion. That's the best way to get a firm grasp on the limitations of any ability, in regards to range, cooldown, &c. One of the best things that ever happened to me was getting a mirror-match in a normal right after I bought Fizz, taught me a damn lot about how he works. Incidentally, now that so much of his Q and W damage has been nerfed, Playful/Trickster is pretty much going to be required for Fizz to get off a good burst now, so that's forcing players to use it to engage rather than to get out after engaging.