Is it permissible to authorize someone to buy toys for me online with his money, and then I pay him back in cash, knowing that there is doubt about the source of this person's money, or any other person in general? And am I obligated to inquire about the source of his livelihood?

The fundamental principle is that people's money belongs to them, and there is no obligation to investigate the source of the money of the person one is dealing with, nor to refrain from dealing with a Muslim whose financial situation is not openly known. Such investigation is a blameworthy over-exertion. A Muslim should avoid unlawful money if he knows it to be so, such as usurped or stolen property, and it is not permissible to take it. However, if the status of the money is unknown, the matter is based on the original principle. If most of a person's money is unlawful, there is a difference of opinion on the matter. As for a Muslim whose financial situation is not openly known, there is no doubt in dealing with him. Even concerning someone proven to have unlawful money, dealing with him is not absolutely forbidden if his money is not entirely unlawful and the transaction is not directly with the unlawful money itself. The Prophet, peace and blessings be upon him, dealt with Jews and polytheists despite their involvement in usury and prohibited practices.