Is it permissible for a woman to observe her 'iddah (waiting period) in her father's house after the death of her husband, knowing that she wears the niqab and her husband's brothers live with her, and the kitchen is shared?

The basic principle is that a woman whose husband has died should observe her iddah (waiting period) in the marital home and should not move from it unless there is a legitimate شرعي excuse. The mere presence of brothers or a shared kitchen does not permit moving to her family's home, and she can take precautions to avoid mixing. However, if she fears suspicion (repletion) or temptation for herself, then it is permissible for her to move.