What is the ruling regarding the Hajj performed by the maternal uncle on behalf of his sick mother, and what are the obligations of her children concerning their mother's vow and the charity calf, especially since the maternal uncle did not adhere to the agreed-upon conditions and there are doubts about his financial integrity?

The one tasked with performing Hajj on behalf of your mother must have already performed Hajj for himself first. If it was stipulated that he perform Hajj by plane, then he is only absolved of his duty by performing it that way, and he must return the money if he violates this condition. Regarding the items she bequeathed to be distributed as charity: if she paid their cost, she is entitled to reclaim it; otherwise, the brother is not obliged to provide them. The sister's vow to sacrifice a calf as charity must be fulfilled by her, and the brother is not obligated to do so. What is obligatory upon the children after the mother's death is to fulfill her vows and perform Hajj on her behalf from her estate, if Hajj was obligatory upon her, or if she bequeathed it and a third of her wealth was sufficient for it. If she did not leave any wealth, then they are not obligated, but their action would be an act of righteousness towards her.