Yea I think Zero destroys him. Wolverine wrecks him pretty bad as well.
Doom isn't completely helpless against Zero, but if Doom presses a button at the wrong time, he's done. He has a hard time mounting an offense, has bad options against Zero's pressure, and trying to zone is just asking to get sniped with lightning. People who complain about Doom's support value are justified in doing so. People who think point Doom is a menace are just terrible at the game.