Does a kaffarah for an oath become binding upon me after I swore on the Mushaf that I would not communicate with anyone except for the service of Islam, if I return to communicating normally, while avoiding messaging girls except in what pleases Allah?

The purpose of the oath is to prevent forbidden communications or messages, or those that lead to what is forbidden. Do not break your oath by making ordinary calls or sending ordinary messages. However, you would be sinning and breaking your oath if you resumed making calls or sending messages that your father swore you to abandon. All of this applies if you are an adult.