How can one reconcile caring for a mother whose father has passed away with appeasing a wife who mistreats her mother-in-law, especially when the wife threatens to resort to the courts?

If your mother needs service, your wife is not religiously obligated to serve her. If she does, it would be good and she would be rewarded for it. Your wife must treat your mother well, out of respect for you. Your travel abroad to earn a living is permissible, provided there is safety and your mother is left in safe keeping. Your wife is not obligated to live with your mother in the same dwelling, as she has the right to an independent home. Filing a lawsuit in court because you advised your wife is impermissible, and marital problems should be resolved through mutual understanding between you.