1. Favicon is already changed on your website https://gyazo.com/f9b6ae04730de1b51cdff020971dc57f

2. By default, this header layout doesn’t have these links. Add the following code snippet to the child theme functions.php file to show them

add_filter( 'basel_header_configuration', 'basel_custom_header_configuration', 1, 1 );

function basel_custom_header_configuration() {

		return array(
			'container' => array(
				'wrapp-header' => array(
					'logo',
					'main_nav',
					'right-column' => array(
						'header_links',
						'search',
						'wishlist',
						'cart',
						'mobile_icon',
					)
				)	
			),
		);
}

3. Change this “Purchase” link in Appearance -> Widgets. Light color scheme for the header is also set for this particular page. You can change it while editing it in the Dashboard.

4. What exactly you failed to do?