OK, I just figured something out. All folders and files located BEFORE the TEMPLATES directory will be used by ZenCart to also affect the layout. The custom folder and template I created will not be the only thing affecting the look of the site.
Not actually correct, if I understand what you meant, If you have created the template "Some_Name" and also edited the template_info.php file for it in it's folder, this will show up in the admin > tools > template selection area where you set this to be your active template..then Zen looks to any "Some_Name" folder for your edited files(only edited files should reside in these folders) and if there is no file there Zen looks to the default location for the file.

Once the template is set, other files, over the ones that you have edited, should have no effect.

If you have created a template or used one from the DL section you should not have a stylesheet_new.css in your "Some_Name" /css folder