How can one deal with a wife who left the marital home after an argument between the husband's and wife's families, and her family is demanding financial guarantees for her return, especially with previous tensions due to the husband's mother's interference in the couple's life and the wife's shortcomings in household matters?

You did well by refusing to give the requested guarantees, as they are not their right. Your wife erred by leaving the house, and you should bring in people of goodness and knowledge to reconcile between you.

If she returns, treat her with gentleness and honor, and keep her away from friction with your family. Your wife's service to your mother is voluntary and not obligatory.

Your performance of household chores is not a defect; rather, it is a required cooperation, and the Prophet, peace be upon him, used to help his family.

Be keen to conceal the details of your marital life. If your wife returns, try to move to an independent home away from your family.

If she refuses to return and you despair of her betterment, then separate from her. You can marry a second wife if your circumstances permit before considering divorce, otherwise, divorce her.