What is the ruling on speaking with one's beloved paternal cousin, whom one wishes to marry, and keeping her pictures while she is wearing hijab, provided that the শরীয়াহ (Islamic law) boundaries are observed?

Your female cousin is a non-mahram to you. Therefore, you must observe the Sharia guidelines in your interactions with her and not transgress the limits set by Allah Almighty. It is not permissible for you to speak with her without necessity, or to keep her picture, even if she is veiled, as all of this can lead to temptation (fitnah). The Sharia came to block the means to evil.