How to add a field on Date Formats setting form?
I wouldn't do this by editing core files. I would implement the buildForm and postProcess hooks in a module extension.

https://wiki.civicrm.org/confluence/display/CRMDOC/Create+a+Module+Extension

https://wiki.civicrm.org/confluence/display/CRMDOC/Hook+Reference

But having said that I probably also wouldn't add it to a core form, I would make a settings page for my extension, which is just a module extension that implements the "add a basic web form" instructions at the first link.