Is communicating online with a committed/pious girl for the purpose of marriage, which includes religious and cultural discussions and encouragement to obey God, considered forbidden if no transgressions occur and hope for marriage is not قطع off?

As for chatting, there is no harm in it if it is through public forums. However, private chats should be approached with caution due to what they might lead to. You must close the door to Satan by deleting her email. As for marrying her, the Prophet, peace and blessings be upon him, said: "There is nothing better for two who love each other than marriage." One should not be deceived by knowing someone through the internet; rather, inquire about her. If she is suitable, seek guidance from Allah (istikharah) and propose to her. If marriage is facilitated, then praise be to Allah. Otherwise, turn your heart away from her, for "...it may be that you dislike a thing and it is good for you; and it may be that you love a thing and it is bad for you. And Allah knows, while you know not." (Al-Baqarah: 216). Ask Allah to replace her with someone better for you.