Marriage proposal

Dear scholars, As-Salamu `alaykum. Can a Muslim woman propose to a Muslim man?

Dear Sheikh,  Asalamualaykum. I have been speaking to a muslima for the purpose of marriage for some time. I am 26 and she is 22 and we are both serious about finding a spouse for marriage. We live in different but neighbouring countries in the West and have gotten to know each other via internet. Considering the significant efforts and expenses required to arran ge a meeting given our respective locations, I would like to have as much information about her as possible and make sure the chances of this working out are good enough before making the trip. My question therefore is: is it permissible for her to show me a photo of her showing her hair if that could encourage me to fly over to where she lives and talk to her and her parents?  She is a hijabi and a hanafi. Jazakhallahkhair.

Asalamulikum, I am interested in finding a woman to marry and would like to know the correct way of going about this step. I understand that it is only permissible to look at specific women only after gathering enough information about the person that convinces one offer a proposal and that one can look without her knowledge. My question is can one look randomly if out of leads? Also, what is the ruling of matrimonial sites(there are some advertised on Islamonline).