Is it permissible for my husband to remain silent about his mother's abuse and her speaking about me in front of neighbors, and his failure to protect my reputation as a wife?

If the wife's mother speaks ill of her, she is sinful, and the husband should gently advise her. The default is to think well of her and not accuse her of backbiting without proof. If neighbors transmit her backbiting, this is forbidden calumny, and the listener's duty is to forbid them, not to inform the one being backbitten about it. The one being backbitten should rebuke the calumniator. It is part of a wife's good character to be kind to her husband's family, overlook their slips, and help him be dutiful to his parents.