Here’s how to get the page you have assigned as the ‘Static Front page’ in the Settings.
$id = get_option( 'page_on_front' );
Here’s the blog page option.
$blog = get_option( 'page_for_posts' );
The current themes name.
$theme = get_option( 'template' );
The website name. You can echo these with bloginfo() rather than returning the values..
$site = get_bloginfo('name');
Website description
$desc = get_bloginfo('description');