Anna lived ex cloister, that is out of the convent. She could dress as she wished, albeit modestly.
She was never tonsured. I do not think most nuns are, they are veiled.
I could be wrong about this, but it is my impression that she entered the order of an Orthodox nun, which for the most part, do not have orders. However I think she had the minor order of St Catherine, as the Empress's friend.. That is just a guess on my part, though.