Changing the colour of featured badge

Have a look at
GD > Settings > Pages > GD Archive Item page > Edit.
You will find this element:

[gd_post_badge key='featured' condition='is_not_empty' badge='FEATURED' bg_color='red' txt_color='#ffffff' css_class='gd-ab-top-left-angle gd-badge-shadow']

Change the bg_color as required.

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000000 is not a valid color, you will need to use #000000.
The shortcode you give is not for the standard featured banner across the image, so if we need to review anything, please post your URL and WP admin details in a private reply so we can have a look for you.

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To show featured first in GD Listings elements, use the “sort by” option.
To show featured first on archive pages, use the Sorting tab of the Custom Post Type.

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If we need to review anything, please post your URL and WP admin details in a private reply so we can have a look for you.

Where are you not seeing featured listings first? URL please?

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Change the sort settings to descending.
On search pages, the results are always ordered by relevance to the search.

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You can add a GD Listings element if you want to the search pages and only show featured listings in that, but visitors expect listings to be ordered based on their search query.

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You can add a GD Listings element (widget or block or shortcode) to the top of the search page and only show featured listings in that if you like.