Is it permissible for a woman to complete her iddah (waiting period) in the home of her deceased husband's family, or in her own family's home, and is it permissible for her to communicate in writing with the deceased husband's brother regarding matters concerning his brother and children?

A woman whose husband has died must observe her 'iddah (waiting period) in the marital home. This is based on what was narrated from the Prophet, peace and blessings be upon him, and the statements of a number of companions and jurists. She is permitted to move from the marital home if there is a valid excuse, such as fear of collapse, drowning, or an enemy, or if she is asked to vacate the house, or if she is unable to afford its rent.

As for speaking with non-mahram men, there is no harm in it if there is a need, whether directly or over the phone, provided that Islamic guidelines are adhered to, such as wearing the hijab, avoiding khalwah (private seclusion), and not softening one's speech. She is not permitted to shake hands with non-mahram men.