Brief Answer:
please meet your doctor
Detailed Answer:
hello!
first thing is that i would like to re assure you that it is not very serious and you dont have to panic!
There can be two reasons for the bleeding
the first and most likely cause is infection of the nose and 
sinuses which accompanied the 
cold has caused the blood vessels to become swollen and this has caused the bleeding
the second and rarer cause will be due to 
cough, causing raised pressure in the blood vessels causing the bleeding.
the 
chest pain is due to the cough and will come down when the cough stops.
You will have to meet your ent doctor to confirm this , he will do an 
endoscopy of your nose and find out the cause for bleeding. It is even better to go to the emergency room now. 
You can ask him to instil 
oxymetazoline nasal drops in his nose , 2 drops , 3-4 times a day , this will control the bleeding.
You can also ask him to take linctus 
codeine 2 teaspoon twice a day for the cough , this will give him relief , 
He may need antibiotics if there is an infection in his nose and i would like you to meet your doctor for the prescription.
You can also ask him to take steam inhalation 3-4 times a day , this will reduce the est and sinus infections.
Take care!