A cross dresser is a person who has unresolved issues from childhood. There are many people that are born both sexes and the parent decides the sex, they make the decision. It happens more often they they let you know and in America it is taboo to be born both sexes but it is an honor in India and other countries. There are those who are infertile and those that are fertile both sexes. That could be one possibility.
The other could be that his hormones could be more feminine in nature and he navigates towards that. He feels more comfort from wearing womens clothing and it helps him. There are many men who in the privacy of their homes wear womens clothing. I think it was stress for him to go outside and pose the way he did. However now the world has said do whatever you want, that is the new normal now and that causes more confusion cause it seems guidelines are getting blurred.
I recommend that you see your dad as one struggling with issues he has a hard time with. That you love him as your dad and forgive him for not thinking of your feelings first and your mothers when he was struggling so hard to keep it together. Have some compassion on him and accept him as he is the way he is. Love your father and just be there for him cause he needs you as much as you need him. (((hugs))) tc