A little bood is normal with a cold that long. Also if your home is very dry due to running heater (this is the case where I live) dry heat can cause nose bleeds. Go to doctor get you some meds a cold does not last 4 weeks you probably have a sinus infection or even pneumonia both of which require antibotics.
no this is not normal...you should take a chest x ray and as well as met a doctor
If you have been blowing your nose too hard you may have ruptured a blood vessel in your nose.
If the blood was mixed with green or yellow mucus, it is likely you have a sinus infection.
The pain in your cheek also indicates you have a sinus problem.
As you have now been coughing for 4 weeks, I would suggest you get examined by your doctor.
Make sure you drink plent of fluids, avoid milk products at the moment because it encourages mucus production.
You can do steam inhalations by pouring hot water into a bowl with some Vick or olbass oil and inhale the steam with a towel placed over your head. Take care not to get scalded.
Best wishes.