I suggest that you write Shin Hae a note. You can explain first that you like her and don't know what the proper way is to ask for a chance to get to know a young woman in her work situation in Korea. If it is not proper for you to ask to learn more about her at all, or if she does not want to know you except as a customer, then you will accept that and not bother her. If she wants to explain why she needs this distance (such as having a future husband already chosen for her) that would help you to understand. But you don't want to offend her or violate her customs--but you don't know what is proper. You hope that writing out her answer to this note, even a very short one, will be easier and more proper than speaking in person.
I suspect that you're facing pretty strict customs, tight family restrictions, and strong embarrassment about doing the wrong thing in public. that's why asking your questions in writing could be easier for her to deal with, because she can take them home and discuss with family members and then write a reply with others' help. I would also caution you that if you actually got so far as to marry a girl whose upbringing and customs are far different than your own, your intimate relationship could be very different because of cultural and languages differences that you would have a very hard time working out-- plus the issue that she might be torn apart inside if she had to choose between being your partner and staying forever very close to her parents. You might be better off finding another person with an adventurous spirit like your own, who has chosen already to move far away from her family in order to become a part of the wider world outside of her own country.