Does a Muslim or a non-Muslim find happiness or worldly recompense for a good deed they performed if it was not done with a sincere intention for the sake of Allah, or is worldly recompense conditioned upon sincerity to Allah and Islam?

Good deeds whose validity does not depend on the intention of worship, such as the example mentioned in the question and similar cases, do not prevent a disbeliever from experiencing their effect in this world. The Messenger of Allah (peace be upon him) said: "Indeed, when a disbeliever performs a good deed, he is fed with it a meal in this world. As for the believer, Allah saves his good deeds for him in the Hereafter and grants him sustenance in this world for his obedience." Scholars have unanimously agreed that a disbeliever who dies upon disbelief will have no reward in the Hereafter and will not be recompensed for any of his deeds in this world. As for the Muslim, he finds the effect of these deeds in this world and the Hereafter, even if he did not intend to draw closer to Allah, given that they are of widespread benefit. If he did intend to draw closer to Allah, his reward is multiplied. Widespread benefit contains good even if one does not intend it, and the proof for this is the noble verse: "There is no good in much of their private conversation, except for whoever enjoins charity or what is just or reconciliation between people," and His saying: "And whoever does that, seeking the approval of Allah, We will grant him a great reward."