I picked the topic of brain freeze because it happens often and to about everyone. But so far has not been known to have a serious long term effect on the brain or body so the only damage you can cause by drinking while having a brain freeze is more pain or a longer brainfreeze.
During this task I learnt that there isn’t exactly one true theory to how a brain freeze is caused. One theory is that you get a brain freeze from extremely cold foods or liquids touching the roof of your mouth because the mouth is very vascularized (hence why temperatures are taken in your mouth). Or (the more fancy version) when you swallow something cold and it hits your carotid artery (back of throat) where blood is fed to your brain and causes a fast change of temperature. Then the cold goes to the receptors surrounding the brain called the meninges causing them to portray it as pain. The last theory I found was that the cold changes the blood flow for a short period of time.