The traditional way is to create a template file for products: node-product.tpl.php. This goes in your theme folder along with all the others. Start with a copy of node.tpl.php and modify it. If you're using product classes, you can use the class name instead of "product" for those node types.

To fix your edit page, try completely uninstalling Contemplate on the modules page. Hopefully that deletes all the files and changes it has made.