Is fixing a car for a sister who works in a mixed-gender environment where Sharia controls are not observed considered cooperation in sin, knowing that not fixing it will lead her to interact with men to fix it herself? And what is the criterion for distinguishing between cooperation in sin and other acts?

The legal intent behind denouncing evil is to eliminate or alleviate it, and denunciation that increases evil is not legally مطلوب (required/demanded). Al-Kharshi mentioned conditions for enjoining good and forbidding evil: that the enjoining party be knowledgeable of good and evil; that they be certain that their denunciation will not lead to a greater evil; that they know or believe that their denunciation will remove the evil; and that their command be effective and beneficial. If the first two conditions are missing, enjoining and forbidding become haram (forbidden). If the third condition is missing, the obligation (wujub) is dropped, and it remains permissible (jawaz) or recommended (nadb). Therefore, if your failure to repair the woman's car will increase the evil of her mixing with men, you should proactively repair it yourself and prevent her from going, as this falls under reducing evil, not aiding it.