Cold enough to freeze the balls off a brass monkey.

This came from a nautical source. the brass monkey was a triangle at the front of a gun boat on which the iron cannon balls were mounted. In very cold weather the brass would contract more than the iron balls which would then roll off the brass frame.

I suppose shaking the balls off a brass monkey came from the same source.

Hair of the dog that bit you.

This comes from the days when rabid dogs were very plentiful. If bitten it was suggested that if you get a hair from the dog and place it across the broken skin there was a good chance that you would avoid getting rabies. I suppose the antibodies in the hair would give some sort of protection.