To what extent is it permissible to admire an unknown person in mixed-gender environments? Is it permissible to speak with him with the intention of a legitimate marital acquaintance? How can one overcome this admiration? And what is the ruling on involuntary gazing?

Scholars have explicitly prohibited speaking with an unfamiliar young woman as a preventative measure to avoid falling into temptation. Therefore, it is not permissible for a young man or woman to speak with non-mahrams, whether in academic settings or through communication channels.

The correct method for someone who desires to propose marriage is to do so through family or trustworthy individuals. If this is not possible, then one must cease thinking about or having any contact.

As for an accidental glance, there is no sin in it if one lowers their gaze and does not continue looking, as narrated in the Hadith by Jarir ibn Abdullah, may Allah be pleased with him.