How can one propose marriage to a foreign girl residing in Turkey, taking into account cultural differences and the difficulty of knowing her family?

If there is no way to propose to a girl except by speaking with her, then there is no objection to that, provided that the legitimate Islamic guidelines are observed. It is permissible for a woman to propose herself for marriage, even if she has guardians. The matters your friend mentioned do not prevent her marriage, but they should be considered and their implications pondered. Piety and good character are the foundation for choosing a spouse, but compatibility between the spouses in other matters is desirable for the closeness of temperaments and ease of understanding.