Is donating a dinar found in a goods cart after the original cart was lost considered valid, and what should the doer of that do?

You must repent to Allah for taking someone else's cart, which caused them to lose their security deposit of one dinar. You must return the dinar to its owner if you can reach him. If that is not possible, then donate it as charity. As Sheikh al-Islam Ibn Taymiyyah mentioned, any money whose owner is unknown should be given as charity and spent on the welfare of Muslims.