What is your advice to men who allow their women to appear before and intermix with non-mahram men under the pretext of custom, or purity of heart, or ignorance of the Sharia ruling?

It is obligatory for a Muslim to present customs to the Shari'ah. What the Shari'ah approves is permissible to do, and what it forbids must be abandoned. The mere habituation of people to something is not proof of its permissibility. If customs involve leniency in being alone with a non-mahram woman or uncovering the face before non-mahrams, then these are invalid customs that must be abandoned, just like habitual adultery or drinking intoxicants. A Muslim woman must veil herself and cover her face and her entire body from non-mahrams. It is not permissible for her to uncover her face or any other part before her paternal cousin, her sister's husband, her husband's brothers, his paternal uncles, or his maternal uncles. She may engage in appropriate speech while observing the hijab and avoiding seclusion, for Allah Almighty says: (And when you ask [his wives] for something, ask them from behind a partition. That is purer for your hearts and their hearts), and He Almighty says: (O Prophet, tell your wives and your daughters and the women of the believers to bring down over themselves [part] of their outer garments). All Muslim women must fear Allah Almighty and beware of what Allah has forbidden them, such as displaying their adornment to anyone other than those to whom Allah has permitted its display.