Am I obligated to ascertain whether the money of the person gifting me is mixed or entirely unlawful?

There is no obligation to investigate the source of wealth of someone you deal with unless you strongly suspect that their money includes unlawful (haram) elements. What is unknown is like what does not exist, and the default assumption is that whatever a Muslim possesses is rightfully theirs. If you are unaware of the proportion of unlawful money a person possesses, then safeguard your religion and honor by avoiding what makes you doubt. It is preferable to abstain from anything that has a strong suspicion. A gift from someone whose wealth is known to be lawful should not be refused. A gift from someone whose wealth is known to be unlawful is forbidden. If there is doubt, it is more cautious to refuse it. The preponderant opinion is that it is permissible, though disliked, to deal with someone whose wealth is mixed (lawful and unlawful).