In Sharepoint 2013, is there a way to override the personal action menu and implement my own customized menu?
Yes, there is a way. Use a delegate control and extend the control responsible for generating the personal action menu, overwriting the resulting menu with your own items. Oh, and by the way, the SE overlords stipulate that I should comment below your original question if I need clarification for what you are needing. They fail to realize that only someone with 50 reputation can do that. So, if you need more information that I've provided please revise your question. And if the overlords want more participation from casual users of this site they will change the 50 reputation policy. It is stupid and counter-productive.