Is the wife sinful if she abstains from intimacy with her husband and avoids him—despite her dislike for him and her desire for separation—due to his bad character, his extreme stinginess towards her, and his refusal to take their children to her mother's house?

It is not permissible for a wife to refuse her husband's bed except for a valid excuse, and what the questioner mentioned is not an excuse. It is also not the husband's right to prevent his children from visiting their grandmother without a legitimate reason. From the perspective of good companionship, a husband should sit with his wife and show her affection. What the wife needs in terms of food and clothing is an obligation upon the husband, and if he falls short in that and the wife finds money belonging to him, it is permissible for her to take from it an amount sufficient for her maintenance. The wife's salary is her exclusive right, and it is part of good companionship for her to assist her husband in the burdens of life. Understanding between spouses, patience, and overlooking mistakes are necessary to preserve the family.