Is it permissible to tell a girl directly, in the street or market, of one's admiration for her and desire to marry her, without referring to her family? And is there any evidence for this from the Qur'an and Sunnah?

It is permissible for a mature woman to propose herself for marriage, as in the hadith of Umm Salamah (may Allah be pleased with her). However, it is better and safer for the religion that a woman be proposed to through her guardian, to close the door to discord and for fear of being exposed to suspicion. Therefore, we advise proposing to her through her guardian. If she only wishes to be proposed to directly, then it is better to have a female mahram of yours act as an intermediary, and that the communication between you be limited to what is necessary and without exceeding what the Shari'ah has permitted, while adhering to the Shari'ah guidelines such as avoiding seclusion and lowering the gaze. We also warn you against letting your gaze wander freely in streets and markets.