Is it possible to have multiple intentions when serving one's children and household, such as fulfilling the needs of Muslims by feeding them, feeding people with the intention of supplicating for them, and assisting the distressed with the intention of supplicating for them? And is there a righteous intention whose reward is greater in raising children and in life in general?

You have done well to include the intention of charity towards family and children, for they are most deserving of goodness, and you will be rewarded for it as charity if you seek the reward. Indeed, deeds are judged by intentions, and whoever is heedless of the intention will receive no reward. The righteous predecessors used to form intentions even in their daily customs, such as feeding those from whom one hoped for the worship of Allah, and raising children to be righteous so they would not harm people.