It would appear that pointy hats were quite common in many parts of the world including Japan. Although the one we usually think of is the 'dunce's' hat given to schoolchildren to wear.https://en.wikipedia.org/wiki/Pointed_hat
As regards the lady's pointy hat apparently they weren't well received by the devout, in France at least, where they were ridiculed by certain people.https://en.wikipedia.org/wiki/Hennin
According to this website the witch's pointy hat is a relatively recent development from the eighteenth century although the article does refer to other possibilities.http://www.slate.com/blogs/the_eye/2013 ... esign.html
From what I remember reading about 'witches' in England they were usually victims of local feuds and the attentions of landgrabbers (property developers in modern parlance).
Although witches could also be male it seems that women were the usual victims. Often old and widowed. What didn't help them was the fact that they were often knowledgeable regarding herbal remedies and in an age where all knowledge was in the teachings of the 'godly' anything to do with curing illness through plants and natural concoctions rather than through prayer would be suspect.
It's interesting to note that even in modern medicine there is a difference in medical titles. Doctors in the past were educated hence the 'doctor' bit. Messy surgery and tooth pulling was left to barbers. Hence the doctor in his or her surgery who talks to a computer for three minutes before printing a prescription out and the Mr or Miss in the Dentist's surgery or hospital operating theatre.