weird theory about melting in weird shapes. Probably it comes from the idea that an ice cube thrown in warm water would break into sharp pieces. Touching the skin would not do that, would just melt to a ball and everything...
The only risk is maybe catching a cold (I am interested in an opposite oppinion since a lot of people think that washing with cold water - mostly for a girl - makes you catch a cold in your genital area and I don't believe it - just never happened to me)
Water freezes at 0 Celsius of course. The fridge seems to drop that temperature a few degrees, probably around -10 (just heared something in a science TV show), however not a temperature that the body cannot handle.
People freeze and eventually get members amputee after long exposure in low temperature (hours in cold wind) so this is absolutely not the case. Anyway wind and water temperature has different effects for your body (you feel ok in 37 Celsius water like your body temperature, but do you feel ok exposed to 37 Celsius air temperature?)
We are talking about water and ice cubes from the fridge. I hope you understand we are not talking about Liquid Nytrogen or anything that would make you an instant statue and break you into sharp pieces.