Is a wife's failure to inform her husband about the loss of her gold, which was in her family's home where the husband had entrusted it for safekeeping, considered a betrayal of trust? And does the father bear responsibility for this loss?

The gold mentioned in the movable property list is owned by the wife, not a trust with you or her father. Neither of them is considered treacherous for not informing you. To clear yourself of any future claim regarding it, you must document in writing, or with witnesses, that the wife has received it or that it was lost while in her possession.