Is the mother’s divorce valid or void, given that she was divorced by court order after claiming she did not know her husband’s whereabouts, and he had abandoned her for ten years? What should she do? And must she observe the iddah if the husband dies, knowing that she did not observe the iddah after the court divorce?

The validity and effectiveness of your mother's divorce depend on the validity or nullity of the court's ruling, and we do not know on what basis the court issued its judgment. A judge's ruling does not make unlawful things lawful nor lawful things unlawful. The waiting period ('iddah) begins from the issuance of an effective divorce. A wife with whom consummation has occurred observes the 'iddah from the time of divorce, even if a long time has passed since she was with her husband.