Would the Hulk be dead because he is a failed super soldier and Zemo shot all of the other known super soldiers. Wouldn't he have shot Bruce too because he is a failed super soldier?
Hulk is not one of the failed Winter Soldiers. He is an attempt by Bruce Banner to recreate the super-soldier serum with Gamma Radiation, which caused his inner anger to manifest as a giant green monster. Hulk had been active for years prior to the events of Captain America: Civil War, so he couldn't have been in the tube to be killed by Zemo. Additionally, Hulk can't be killed with a single bullet to the brain. Banner tried that, and Hulk spat out the bullet.
Multiple reasons really:
Hulk was not one of the Winter Soldiers
Hulk wasn't a Winter Soldier but rather a creation of his own. As such he wasn't in the chamber where Zemo killed the Winter Soldiers in Captain America: Civil War.
Helmut Zemo: If it's any comfort, they died in their sleep. Did you really think I wanted more of you?
Captain America: Civil War
Banner was MIA
At the time of Captain America: Civil War, when Zemo surfaced, Banner/Hulk were off world. No one knew where they were until Thor accidentally stumbles into him in Thor: Ragnarok. Hard to kill someone if you can't find them.
Hulk isn't exactly killable
Hulk has survived a lot, Thanos, Thor, Fenris, pretty much every broken bone, bullets as Banner, bullets as Hulk. He's not exactly the sort you can just kill.
Bruce Banner: In case you needed to kill me, but you can't! I know! I tried! I got low. I didn't see an end, so I put a bullet in my mouth and the other guy spit it out! So I moved on. I focused on helping other people. I was good, until you dragged me back into this freak show and put everyone here at risk!
Is it well known?
Hulk isn't exactly a normal super soldier. He's a mutated form. He doesn't really scream "super soldier" when you look at him. On top of that Banner didn't even know what he was working on. It's possible it wasn't really common knowledge that outside of S.H.I.E.L.D. that Banner was working on the formula and so that's essentially what he was.