Depends. What are you ill with? If it is a simple cold it should be fine. If it is something more serious then maybe not. You could call your doctor for a final opinion since your doctor knows you and your condition better than anyone else.
Maybe. You don't specify the illness. If it's a simple cold, there really isn't any reason not to do it. A more serious illness would require consultation between the dentist and your physician to determine the advisability of the procedure.
It is possible. So, an extraction will tax your energy level and you will need to recover from it. It could prolong your overall recovery, but if you are up to it - it will not significantly affect your recovery. I always tell patients it is their call.