If you have a fever, I would definitely not recommend running. However, if there is no fever and you feel up for the run, it is probably okay to run. It would be the most safe if you decreased the intensity and duration from normal. But don’t be afraid to let yourself rest while you are under the weather. It will go away and then you can get back to your workouts. I commend you on your motivation!
Yes, fever it definitely out. But I would also advise against it druing the few worst days of the cold. If you feel that you must to adhere to an exercise schedule, try a different modality that does not place such a stress on your cardio-respiratory system. If you are a walking germ-bomb, consider using an exercise video at home. Once you can easily take deep breaths, by all means go out an run again.
That depends on your symptoms. If you have access to one of those 24 hour nurse hotlines, I would ask, but the choice is really up to you. I have exercised during sinus infections but took it easy and modified intensity and length, including running. Above all, just make sure you are taking steps to get healthy and it’s okay to let your body recover from a cold or illness before resuming your fitness program.