What is the reward for schizophrenia patients in this world and the hereafter? Are they considered "mabṭūn" martyrs if they die from their illness without committing suicide? And will they be forgiven if they were not observant in prayer but were Muslims, given their disability in working and their hallucinations?

If schizophrenia affects the mind, and the afflicted person loses their mind because of it, then they fall under the ruling of an insane or imbecilic person. They are not obligated to perform prayers, nor to make up for any missed ones. However, if they occasionally regain lucidity and are aware of the prayer times, then they must make up those prayers. If they do not lose their mind, then they are obligated [to pray] like other sick individuals.

This disease is not an abdominal illness whose sufferer is considered a martyr. However, the afflicted person is rewarded for their patience with it, as the reward is commensurate with the severity of the illness.

There is no linguistic difference between the words "balāʾ" (بلا) and "ibtilaʾ" (ابتلاء); both mean testing. If the questioner's intent is whether it is a test or a punishment: if a person is born with it or it afflicts them before puberty, then it is an "ibtilaʾ" (trial) and a test. If it occurs after puberty, it could be a "balāʾ" (affliction) and a punishment, or an "ibtilaʾ" (trial) and a test. In all cases, the believer is rewarded: if prosperity befalls them, they are grateful, which is good for them; and if adversity befalls them, they are patient, which is good for them.