Is it permissible, in Sharia, to send a picture of one's passport to a girl to identify oneself, and is it permissible for the Sharia-sanctioned viewing (Al-Ru'yah Al-Shar'iyyah) to take place through conversation via Skype in the presence of her guardian?

It is permissible to speak with a non-mahram woman within the Islamic legal guidelines if there is a valid need, and not merely for the sake of acquaintance. As for engagement and seeking marriage, these are considered valid needs. It is permissible to send her a picture of the passport, just as it is permissible to perform the legally sanctioned viewing (ru'yah shar'iyyah) via electronic means, provided six conditions are met: the strong likelihood of her acceptance, after the decision to marry and before the engagement, not repeatedly looking except as much as needed, looking only at the face and hands, safety from lust, and that it is not with pleasure. It is preferable for the guardian (wali) or a mahram to be present.