It is normal to have some minor reaction to the eating of very cold foods. Typically, this reaction is what you describe as 'brain freeze', characterized by sharp but brief pains in the sinuses or teeth. This reaction is not caused by a serious underlying problem, and it is usually experienced by most people from time to time.
However, you definitely should not feel like your throat is closing up when you eat something cold, nor that it eating something cold makes you need to cough. These symptoms that you are experiencing could be a sign of a serious reaction in your body, and you need to get it checked out. In the meantime, you should absolutely avoid eating cold foods that provoke the symptoms until your doctor
determines that it is safe to do so.
In particular, in some people exposure to cold temperatures can provoke an allergic type reaction. It is possible that the sensation of the throat closing that you feel when you eat cold feeds represents a localized form of this reaction in the throat. This could be quite dangerous, as severer throat swelling could lead to trouble breathing.
See your doctor for advice as soon as you can.