If posts category is “cars”, display image

You can make use of the conditional tag has_catgeory. You can do something like this

if(has_category( 'cars' ) || has_category( 'uncategorized' )) {
    //display cars image
}elseif(has_category( 'banana' )) {
    //display banana image
}

From your comments, here is your code

<?php if(has_category( 'Update' ) || has_category( 'uncategorized' )) { ?>
    <img alt="Hello!" src="https://wordpress.stackexchange.com/questions/161368/<?php bloginfo("template_directory'); ?>/img/loud.png"> 
<?php } elseif(has_category( 'Event' )) { ?>
    <img alt="Hello!" src="https://wordpress.stackexchange.com/questions/161368/<?php bloginfo("template_directory'); ?>/img/event.png"> 
<?php } ?>

EDIT

The conditional tag in_category can also be used for this purpose

EDIT 2

For the second part of your question, check out this post by @JohannesPille on SO. Here is the post reposted.

PLEASE NOTE: I did upvote the answer on SO. If anyone see this as unfit, please rollback this edit 🙂

The action create_category runs when a new category is created.

You want your category creation function to run when the theme is activated. The relevant action is after_setup_theme.

Drop this in your theme’s functions.php and you should be good to go:

function create_my_cat () {
    if (file_exists (ABSPATH.'/wp-admin/includes/taxonomy.php')) {
        require_once (ABSPATH.'/wp-admin/includes/taxonomy.php'); 
        if ( ! get_cat_ID( 'Testimonials' ) ) {
            wp_create_category( 'Testimonials' );
        }
    }
}
add_action ( 'after_setup_theme', 'create_my_cat' );