Is a gift considered unlawful if there is strong doubt, reaching certainty, that its source is illegitimate (stolen or purchased with a stolen credit card), and what should the recipient do with it if the giver resides in another distant country?

If you know that the money is stolen, then the gift is void, and you must return the stolen money to its owners. If you are unable to do so, then return it to the thief if you know that he will restore it. If all of that is not possible, then donate its value in the path of Allah.