The SEO spider is thought to find issues on a website.

1. You could have an issue or a third party plugin overriding it somewhere but i can't guess the exact reason. When 'Include Site Name in Page Titles' in the Joomla global configuration is enabled normally the site name is always added to all pages

2. The SEO spider should be used with the Metainfo Dashboard that allows you to specify custom metadata and fix duplicates too. I recommend using that being the easiest way to deal with metadata. If you define metadata under Joomla menu and you don't get it, it means that something doesn't work fine somewhere.

Best Regards

John