I want to know is it possible to remove the password changing fields from the Core Joomla profile editor? I want to make a separate page for Password Changing stuff for the front-end users?
You can quite easily override how the core pages are displayed using your templates folder. See How to override the output from the Joomla! core for details. In short, you make a copy of the core form in your template/html folder and modify that copy to your liking. If Joomla finds the form there, it will use that one instead of the installed version.