Prevent specified posts from being featured on the front page or in feeds, and from notifying external services of publication.
This is probably the kind of thing that you would recognize the need for or you don't. It's beneficial in instances where you want to publish new content without any fanfare and just want the post added to archive and category pages and its own permalink page.
Yes. See the Filters section (under Other Notes) and look for the example using the 'c2c_stealth_publish_default' filter. You'll have to put that code into a site-specific plugin or your active theme's functions.php file.
The checkbox is always present since it continues to have an effect on published posts, such as preventing the post from appearing on the front page or in feeds. You may, after publication, decide to not have the post be stealthy. In such a case, you can do so directly by editing the post and unchecking the checkbox; you do not need to change it back to a draft and then republish it.
(This is a developer-level question that doesn't affect general users.) The plugin assigns a custom field of "_stealth-publish" with a value of "1" to indicate posts that are stealth published. Unless, of course, the name of the custom field was changed via use of the 'c2c_stealth_publish_meta_key' filter.