Reply To: "Featured" and "New" flags

You could change the css to show your image – firebug shows me that the “new” image is declared in /geodirectory-assets/css/style.css on line 300, so you can just overwrite this in your child-theme’s stylesheet:

#geodir_wrapper ul li .geodir_new_listing {
    background: url("../images/my-new-new.png") no-repeat scroll left top rgba(0, 0, 0, 0);

I don’t have featured on my test site, but it would be the same principal.