Is it permissible for the aforementioned woman to live in her house with her husband's mother, or her sister, or a righteous woman, and is this considered an unlawful khalwa (seclusion)? And is it permissible for a man to take a woman as a "sister in Islam," and is this established or sanctioned in Islam?

Friendship between a man and an unrelated woman, and proximity to her, is a means to fall into sin (haram (definition)). The Prophet, peace and blessings be upon him, warned against the temptation of women and worldly life. Islamic law blocks all avenues that lead to what is forbidden (haram). It has been narrated from the Hanbalis that it is permissible for a man to greet a young woman to avert corruption. Listening to the voice of an unrelated woman by an unrelated man, and vice-versa, is a share of adultery for which its perpetrator will be punished. Safety is unparalleled.