What is the ruling on thanking a non-mahram woman for a favor she rendered, and is it permissible to apologize to her for speaking with her by saying: "I am sorry, but I do not speak with women" if there is a concern of fitnah, and what is the ruling on returning and initiating salutations to her according to the Shafi'i school?

It is permissible to speak with a non-mahram woman when there is a need, provided that there is no fear of temptation; otherwise, it is not permissible. Thanking a woman when there is fear of temptation is not permissible. There is no harm in apologizing when there is no fear of temptation. The Shafi'is held that it is forbidden for a young woman to initiate greetings with a non-mahram man, and likewise for her to return his greeting. They also disliked a man initiating greetings with her, because her returning or initiating a greeting might lead him to desire her.