ikaring

Ah, that make sense.
To make it crisp in your environment, you need to upload double sized images, and theme setting to use them instead of 960x420px ones.

However, that might affect non-retina displays with too big images. Or might not because of responsive images feature introduced in WP4.4.
I dont fully understand the feature, so pls check by yourself.

Responsive Images in WordPress 4.4

I dont think ppi affects resizing quality.
As to file type, resized quality might slightly differ, I guess. But can’t tell which is good.
It depends on the image.

BTW you can set compression quality for jpg. Pls check this article.
http://www.wpbeginner.com/wp-tutorials/how-to-increase-or-decrease-wordpress-jpeg-image-compression/

Or you can set display: none; instead of margin-top: 0;.
That’s gonna be easier.

However, pls note that updating the theme will delete your customizations.
To avoid that, it is recommended to use child theme.
See Codex:
https://codex.wordpress.org/Child_Themes

After following the steps in the page, and suppose you have style.css and functions.php in child theme directory, one more file is required for removing <header class=”entry-header”>.
That is content-page.php.
So duplicate content-page.php from your parent theme directory ( topcat-lite right? ) to the new child theme directory.
Remove <header> block.

Hi.
MixItUp looks good to me, too.

I saw your page today, and it seems you are almost there.
Since WordPress uses jQuery by default, you do not load another jQuery yourself.
No need to add this line.
<script src="http://code.jquery.com/jquery-1.11.0.min.js"></script>

Actually it is better enqueue scripts with wp_enqueue_scripts function in functions.php, but let’s leave it for now and make filtering function first.
( Just for your reference )
https://developer.wordpress.org/reference/functions/wp_enqueue_script/

Now you have filter buttons like so;
<button class="filter" data-filter=".large">Large Spaces</button>

Set ‘large’ class for <div> of each venue with large spaces, like so;
<div class="mix large">
‘caterer’ class for venue with in-house catering.
<div class="mix caterer">
You can use multiple classes as you need.
<div class="mix large caterer outdoor">

That’s it!
However, I think you need to add those classes somehow using tag or custom taxonomy or maybe custom field.

How about using one of these sorting/filtering jQuery plugins.

https://www.sitepoint.com/top-5-jquery-filter-sort-plugins/

Please check to see if it works when just putting the above code in text editor.

If not, check the correct path to updatepage.js.
If the path is correct, you can use shortcode to add javascript even in visual editor, like [updatepate].

Add this code in your functions.php.

function print_updatepage_js() {
$str =  '<script type="text/javascript" src="/scripts/updatepage.js"></script><script type="text/javascript"> updatepage(); </script>';
return $str;
}
add_shortcode('updatepage', 'print_updatepage_js');

Which theme do you use?
It mostly depends on theme templates.