ikaring

I suppose you understand template hierarchy.
https://developer.wordpress.org/themes/basics/template-hierarchy/

Since there is not category.php nor archive.php, WordPress use index.php for displaying category archive page.
However, index.php is user for other pages like top page, tag pages and 404 page, if there are not appropreate template.
And because contents need to be different in each page, there are conditional statement in index.php, like if($is_404 == true) or if(is_category()).

Now, provided that you prepared child theme, copy index.php of parent theme to child theme directory, and rename it to category.php.
This category.php is gonna be specific template for category archive pages only, so you don’t need if($is_404 == true) block or if(is_category()) wrap.
( You can delete line 2-4, too.

/*
  Template Name: Home page
 */

These lines are kind of odd to be here, but anyway.)

As for code I was talking to add after line 198, that can be anything you want.
If you want to show category description, just paste the following:

<?php echo category_description(); ?>

You are welcome.

There is an error remains on scrolling-nav.js.

Just wrap whole code inside the following:

(function($){
//paste original code here
})(jQuery);

This should fix it.

Hi JakeCr8Guru.
It seems <?php wp_footer(); ?>` is not set before your footer scripts.
wp_footer() handles enqueued scripts for footer area, and it is usually set right before closing body tag.

However, you need those plugin js before your inline footer scripts, so put <?php wp_footer(); ?> before script tags.

It is written in index.php.
Loop starts at line 181.

If you want to add content after thumbnails, you can add code after line 198.

However, it is recommended to make Child theme to customize theme.
https://codex.wordpress.org/Child_Themes

Slider can be enabled or disabled by checking checkbox, which is in Appearance > Inkness Settings > Slider Settings tab.

If above checkbox is checked, nivo.slider.js is enqueued in functions.php around line 133.

content-home.php is called in home.php line 32:
get_template_part( 'content', 'home' );

This command search for content-home.php first, and if none, search for content.php next.

Yes, terminology is a bit complicated.
Let’s check them out.

If you find ‘tribe_events_cat’ menu under ‘tribe_events'( or Tribe Events or some sort) menu in admin side menus, ‘tribe_events_cat is custom taxonomy.

If ‘Conference’ and ‘security’ are in Posts > Categories, they are categories.

If you don’t find them in above place, and you set values for them in each post editting page, they might be a custom field values.

Same for ‘eventDisplay’.

Here are questions:

1. Which type are they? Custom taxonomy, Category, or custom field?

2. Which template file your code is in?

3. Is tribe_events_cat hierarchical? ( Is ‘Conference’ the parent of ‘security’, or they are siblings?)

Sorry, you need different size for thumbnails.

<?php if ( have_posts()) : ?>
	<?php while ( have_posts()) : the_post(); ?>
		<div>
		<?php if (is_first_two_posts()) {
			// for big thumbnails
		} else {
			// for small thumbnails
		}
		?>
		</div>
	<?php endwhile; ?>
<?php else : ?>
<?php endif; ?>

Try the following.
Presuming eventDisplay and tribe_events_cat are custom taxonomy.
If they are custom field values, you need to use custom field parameters: See Codex http://codex.wordpress.org/Function_Reference/WP_Query#Custom_Field_Parameters

Updated: in case you have posts in both security and Conference categories.

<?php
$args = array(
	'post_type' => 'tribe_events',
	'posts_per_page' => -1,
	'tax_query' => array(
		'relation' => 'AND',
		array(
			'taxonomy' => 'eventDisplay',
			'field'    => 'slug',
			'terms'    => array( 'past' ),
		),
		array(
				'relation' => 'AND',
				array(
					'taxonomy' => 'tribe_events_cat',
					'field'    => 'slug',
					'terms'    => array( 'security' ),
				),
				array(
					'taxonomy' => 'tribe_events_cat',
					'field'    => 'slug',
					'terms'    => array( 'Conference' ),
					'operator' => 'NOT IN',
				),
		),
	),
);
$wp_query = new WP_Query( $args );
?>