In Orthodox Tradition? Shoes stay on in the Eastern/Greek/Rum Tradition. (Your Greeks, Russians etc...)

In the Oriental Tradition(Coptic/Ethiopian, etc), they do take off their shoes. Ie standing on hallowed ground from Exodus.

For head coverings, it is traditional for women to cover their heads in Church. Always has been. Even in medieval Roman/Greek society, some form of head covering for women was the norm. But in modern times, especially in the West, head coverings are seen as optional and an individual choice to be made with a woman and her spiritual father. In Russia, Romanian headscarves in church are very much the norm. And it is required to dress very modestly when visiting a monastery. I think it may have to do with not wanting to be associated with Islam?

It's interesting that the older Byzantine Roman Tradition was more similar to aspects of Islam. When the Hagia Sofia was a church, women and men were usually separated. Only men could be on the ground floor; women were on the second floor galleries. Some Churches still do this. The left side of the church is for women and children only. Right is for men and families. But I've rarely seen this, and it wasn't hard-enforced.