Dan Bobinski's Answer:
This kind of situation is certainly frustrating for an employer! Your issues here are twofold. First, the legal ones. Second, the employee's performance.
In terms of legal, the Americans with Disabilities Act prevents you from considering the health of an applicant during the interview process. I don't need the hate mail, but many employers are frustrated with not being able to ask questions that would identify an applicant who might not be able to perform the full scope of the duties required for a job.
This is one of the reasons I strongly advocate having a systematic interview process with metrics related to key job-related factors (my e-book on hiring great employees gives instructions for how to do this).
Always check with your attorney, but the ADA is pretty clear that an applicant is not required to disclose pre-existing conditions, and by law you are not allowed to inquire about them.
With regard to this individual taking excessive emergency time off, if no restrictions or limitations exist in an employee manual, your actions may be limited. Again, check with your attorney as every state law is different, but this is a good reason to have a clearly-written section in your manual or employee handbook outlining rules about time off.
If you don't currently have a manual or handbook, now would be a good time to develop one. Whatever you have outlined in your employee handbook is legally defensible so long as it does not contradict federal, state, or local laws.
Keep in mind you cannot make it retroactive, nor can you focus it only the actions of one individual. But after the rules have been made clear to the employee, you'll have legal grounds for taking action if he's in violation of your policies.
When you say you're told he "often sleeps while on duty," but is clever enough not to be seen, how do we know he's sleeping? The proof has to be in the pudding. If nobody has seen him sleeping, how do we know? If someone were to videotape him in the act, then you have something to work with, but even then, you may need to contend with surveillance laws.
The bottom line? Your only real defensible action is to focus on his performance. Not knowing all the details, it's hard to give specific advice, but you might consider putting this individual on a performance improvement plan, spelling out exactly what is expected in his work behavior. Then, if he doesn't meet those expectations, appropriate discipline may be enacted.
Emergency time off and medical issues must be addressed in accordance with your employee manual and with the advice of your attorney. But performance issues are squarely in your circle of influence. One thing is for sure: You need to do something and be consistent with it or it's going to affect the morale and production of the rest of your employees.