Search

I don’t understand what you are trying to search.
The only searches there are search all WP posts and pages and GD listings [just add

/?s=

after the URL of your site].
Or the GD search, searching all listings: https://wpgeodirectory.com/docs/gd-search

If your logos are not part of posts or pages or listings, you cannot search them.

Thanks

I do not know of any way of searching on a page like that, except for using your browser’s search function like CTRL + F.

There is only one featured image, which is the image of the first result. You could hide or customize that by using this CSS, adjust as required:

.search-results .featured-area {
    display: none;
}

Thanks

Try adding this:
.type-page .featured-area {
display: inherit;
}

If that fails, post your WP admin details in a private reply so we can test better.

Thanks