People with disabilities work in all sorts of jobs. I guess it would depend on what the disability is. Now if you are a sociopath or have problems with psychosis then I would say that nursing is not for you, because first you have to care about your patients and second you have to be rooted in reality 100% of the time when you are on duty.
But there are loads of nurses who function well (with meds, therapy etc) who have a history of depression or manic-depression. I would suggest that you (or whoever you are asking for) talk to their mental health provider and see what they think.
Depends upon the type of disability and the severity with which it limits your functioning. If the problem severely limits your ability to perform your job as a nurse or make sound judgement calls as a nurse, you probably can not do it. Likewise, the reverse. Between you and your doctor to discuss the ramifications of your problem as it pertains to nursing as a career choice.
I don't think anyone would know you had the disability unless it was not controlled. That won't be pulled in background investigation. As long as it's under control, and you don't have breaks from reality and your doctor thinks it's OK, you should be fine.