Is it normal to be hyper sensitive to noises in workplace enough that it does affect performance and mental health or should the person just move on?

No defined normal: I don't believe that there is a defined normal here. You can check with an ENT doctor to see if you have anything wrong, such as hyperacusis, misophonia (wikipedia these to see it they apply), otherwise, it is going to basically be your call. You can see a psychiatrist if the ENT doc thinks misophonia. Ssri's can help, esp.Since you sound depressed. Trotzdem wuensche ihnen frohe weihnachten!

Is workplace safe?: It is not normal to have such hypersensitivity and ill-effects. Is there something about the workplace which is toxic? Do others have similar reactions? If this sensitivity is unique to you then your perceptions and reactions may be hyper-acute due to chemical imbalance. Seek a psychiatrist to help determine which medication can ease the symptoms or else it may follow you as you move on.