Is money earned from an internet cafe, which a customer frequented with his girlfriend, considered unlawful (haram), and does this money mix with the lawful (halal), especially since a portion of it was used to purchase PlayStation games?

Opening an internet café is permissible with Sharia-compliant controls. If it is not controlled, then it is not permissible, because the use of the internet for unlawful acts makes the owner of the café an assistant to sin. Allah Almighty says: (And cooperate in righteousness and piety, but do not cooperate in sin and aggression). It is not permissible to take payment for unlawful use, and if this occurs, the unlawful money must be disposed of by spending it on Muslim welfare and good deeds.