Getting a cut should not be a problem, unless you have a clotting problem. As long as you clean it, and then put a plaster on to the wound to prevent infection/cross contamination, you should be fine. Yes, it takes a little longer to heal, but no big deal really. Unless you have the cut on your feet. Then you need to take care as that can lead to other nasties.
Being ill is a bind, but if you test more frequently and know WHY your blood sugars are all over the place and increase the insulin, then that should not be too much of a problem either. We all get sick, diabetic or not!