What is the ruling on reciting Al-Fatihah with the intention of seeking success for an online engagement, and is it permissible for the father to propose on behalf of his daughter?

You can authorize your father or someone else to represent you in the marriage proposal (khitbah) and the marriage contract (aqd an-nikah). It is preferable to authorize someone else, due to the difference of opinion regarding one person undertaking both parties of the contract. Reciting Al-Fatiha at the time of the marriage proposal has no basis in Islamic law; it is an innovation (bid'ah) from which one must beware.