Easy. As an AP mid laner, you build Zhonyas. Now, this means you can't get your precious mana regen item right off the bat so you have to be a bit more conservative with your spells, shocker. However that's the price you pay for negating his ult and reducing his damage. If you're an AD mid laner, like Talon, then you should win. Talon naturally defeats Zed should Zed not successfully poke him down. If you play something like Yasuo, it's all about dash harassing whenever possible and windwalling his shurikens.
You can't help roams. If he goes down and gets a double kill, that's bot lane's fault. They need to ward, you need to call MIA. That simple. That's not a basis to judge a champion on. You can do the exact same thing.
Don't listen to anyone that claims rushing Zhonyas destroys a mage's early game or something. It's just a price. It's not fair that you can build an item that negates damage a bit, makes his ult do nothing, AND keeps your spell spam. You give up some mana regen in return for him losing a bunch of damage and the loss of ability to tower dive you really.
And yes, you do build QSS or zhonyas on almost everyone with a zed in game. Same as you build a QSS with a WW or malz in game. One could say "Not fair I have to build an item with X champion in game" but the counterplay is that the item just about negates a button on their spell rotation.
Vs Zed, play stuff like Talon, Malz, LeBlanc, Cho Gath.
Make sure you rake him more than he Q's you in lane, auto attack when he tries and CS if you think he won't destroy you. If you do that, once you hit 6 you're unkillable. The second he ults you, you ult and just wait it out and jump on him. Guaranteed win.
Malz requires timing in that as zed ults, you place your null zone under you, pre-emptively place your silence so it'll hit him right when he comes out of his ult, Space aids him, and then ult. Guaranteed kill, especially with ignite.
LeBlanc can burst suddenly, and has so much mobility that if Zed ults, unless she has no health, can often take no damage. Especially since you can jump on Zed and chunk lots of health, and upon falling before a certain threshold many Zed's instinctively ult and upon that, you just port back to your W's starting location and you traded a 12 sec CD or so for a 70 sec CD. And you have mimic so can do it all again.
Cho is just tanky. He is constantly sustaining, eating minions for more health, building RoA. Zed ults, place your Q before he lands, knock up right as he lands. Silence once he comes down from the knock up. Take a bite. If you have ignite, Zed is dead. If not, he's probably still dead. Oh, and zhonyas on any of the AP champions mentioned above. Kills his damage and his ult.