Getting the permalink to the latest post from a category

This will work:

$latest_post = get_post( array( 'cat' => YOUR_CAT_ID, 'posts_per_page' => 1) )
if( $latest_post ) {
    echo get_permalink( $latest_post->ID );
}

Or you can do it in bulk for more categories:

$categories = array( 7, 12, 14, 15 );

foreach( $categories as $cat ) {
    $latest_post = get_post( array( 'cat' => $cat, 'posts_per_page' => 1) )
    if( $latest_post ) {
        echo get_permalink( $latest_post->ID );
    }
}