How can one atone for taking public money that has no specific owner? Is it possible to return the money from installments of a car that was sold after being bought with money mixed with my own? And does unlawful money become mixed with lawful money if it is mixed with my personal funds?

The expiation for ill-gotten gains (haram wealth) is through sincere repentance, which includes remorse, cessation, and a firm resolve not to return to the sin. One must clear their conscience by returning the money to its public authority (the state treasury), even via a postal order. If this is impossible, or if it is known that they will not use it properly, then it must be spent on public Muslim interests or given to the poor and needy. It is permissible to return the money in installments from the sale of the car if the questioner does not possess enough to pay it all at once; otherwise, they must hasten its return. Lawful personal money does not become unlawful by mixing with ill-gotten gains if its owner repents and disposes of the unlawful money.