What is the ruling on speaking with a girl via email for the purpose of getting to know each other before engagement, if her guardian approves of it, and is it permissible to get to know her in this manner with the aim of marriage, while being careful not to fall into temptation?

The temptation of women is great, and the noble Sunnah has warned against it. Islamic law has set a strong fence regarding interaction with non-mahram women. A Muslim's conversation with a non-mahram woman is not permissible except out of necessity, and only to the extent of that necessity, while adhering to the regulations of Sharia. Conversing with a girl under the pretext of wanting to propose or marry is not permissible in Sharia, because it leads to temptation and following the footsteps of Satan. You must cut off all ties with this girl if you desire safety for your religion and your honor. The guardian's approval of the conversation has no legal validity in Sharia. Piety requires avoiding disobedience to Allah Almighty, and a pure intention does not improve an evil deed.