The website you quoted was created with Showkase-Pro and the Juicebox galleries within the site use certain Pro configuration options which are not available to Juicebox-Lite (the free version. However, you would certainly be able to get a close approximation to these galleries with Juicebox-Lite, though (for example a single row of thumbnails below the main image and a unique caption for each image is certainly possible with Juicebox-Lite).

To see what can be done with Juicebox-Lite and Juicebox-Pro, we have several demo galleries here.
You can download Juicebox-Lite from the download page.

Unfortunately, we do not have a trial version of Juicebox-Pro but we do offer a money-back guarantee. Please see this FAQ for details:
Can I try a trial version of Juicebox-Pro?

With Juicebox, you would be able to create as many individual galleries as you like but you might need to manually embed them into web pages following the instructions here.

If you are looking for an automated solution to creating and listing multiple galleries or a way to create a complete website, then Showkase would be more suitable.

Showkase is a PHP web application which allows you to create a complete portfolio website (integrating multiple galleries) online.
Showkase has full support for Juicebox-Pro (and SimpleViewer-Pro) galleries and the galleries can be created within the application itself in a web browser interface (or created with JuiceboxBuilder and imported).

Showkase can also create non-gallery pages (About, Basic and Contact pages) where you can add information about yourself (or any other content you like).

Showkase can be purchased as Showkase-Standard (which comes with Juicebox-Lite and SimpleViewer-Standard, the free versions) or Showkase-Pro (which comes with Juicebox-Pro and SimpleViewer-Pro).
The only difference between Showkase-Standard and Showkase-Pro is the bundled viewers.

More information on the differences between Juicebox and Showkase can be found in this forum post.

Hopefully this will answer any queries you may have. If you have any further questions, just let me know and I'll do my best to answer them for you.

When using Showkase, creating and editing pages and embedding galleries is all done automatically within the Showkase interface without the need for any manual coding at all.

Like Juicebox-Pro, there is no trial version of Showkase but, again, we offer a money-back guarantee (as noted in the FAQ below).
Can I try a trial version of Showkase?

As well as the demo Showkase sites on our own site, we have links to other sites created by Showkase users on our Facebook page, in our Twitter feed and in this blog post.

Here are a few things to try. They may not be ideal solutions but they might help to track down the root of the problem.

(1) Try using the suggestion in the Using Percentage Heights support section on your <div id="photo"> container (the gallery's parent container).

(2) Try giving your gallery a fixed pixel height such as 600px instead of a percentage height.

galleryHeight: '600',

(3) Try using a different imageTransitionType to see if the problem happens with transitions other than CROSS_FADE. A short description of the imageTransitionType configuration option and all its possible values can be found in the Main Image section of the Config Options page.

(4) Try viewing the gallery on a page of its own (using the gallery's own 'index.html' page) to see if the problem has anything to do with the code on the embedding page.

(5) Try adding the following line of code to the .htaccess file in your root directory to ensure that browsers are able to cache the image files for your gallery.

Header set Cache-Control "max-age=2592000"

The results of trying these suggestions should hopefully help with troubleshooting the issue.

In a nutshell, you are unable to upload the 'juicebox.js' file to your web server because your web host scans all files with ClamAV and ClamAV thinks (incorrectly) that the file contains malware and immediately deletes it.
Without the 'juicebox.js' file in place, the gallery will not work.

I have done all I can (by filing false positive reports to ClamAV) and hope that ClamAV will exclude the Lite and Pro 'juicebox.js' files from being flagged as malware in a database update very soon.

Unfortunately, unless ClamAV contact me directly (they have my email address) I have no way of knowing if or when this has been done.
Also, without being able to reproduce the problem, I cannot tell if it has been fixed.
I do not have access to a web server that uses ClamAV and the online scanners that use ClamAV seem to give the 'juicebox.js' files a clean bill of health.

Being that your web host updates their ClamAV database several times a day, the best I can suggest is that you periodically try to upload your 'juicebox.js' file to your web server and see if it works. (It should take less than a minute to check.)

If or when I hear anything back from ClamAV, I will be sure to post back in this thread.

From your screencast video, your gallery settings look OK (although there should be no need to explicitly set imageScaleMode="SCALE_DOWN" as this is the default value for this option)
Without actually being able to see your gallery live on your web server, my best guess as to the cause of the problem is that some CSS code (perhaps from your theme) may be affecting your gallery. For example, if there is some global CSS somewhere on your page which is being applied to all images on the page, then it will affect the images in the gallery, too. You might need to track down this CSS and ensure that it is applied to only these elements on your web page that actually require it.

To see if this is the case, try temporarily reverting to a default WordPress theme (such as Twenty Sixteen) to see if this solves the problem.
If it does, then you could go back to your current theme and check, using your browser's developer tools (usually accessible via the F12 key), to see what CSS is being applied to the images in the gallery.

It is not possible to isolate a Juicebox gallery (or any other HTML element) from global CSS and the gallery has no option but to inherit such rules.
The best course of action would be to have CSS rules applied to only those elements on your web page which require them through use of further CSS selectors (classes and ids).

I hope this points you in the right direction.

If you continue to experience difficulties, then please post the URL to your gallery's web page so that I can take a look at the problem for myself and hopefully help further. Thank you.

If you want to display two divs side by side, you could wrap then in a container div and float one of them left and the other one right.

To see this in action, create two sample galleries with JuiceboxBuilder-Lite.
Save one gallery in a folder named 'gallery1' and the other in a folder named 'gallery2'.
Now create a new HTML file with the code below, name it whatever you like, place it in the same directory as the two gallery folders and open it in a browser.

<!DOCTYPE html>
<html lang="en">
<head>
    <title>Juicebox-Pro Gallery</title>
    <meta charset="utf-8" />
    <meta name="viewport" id="jb-viewport" content="width=device-width, initial-scale=1.0, minimum-scale=1, maximum-scale=1, user-scalable=0" />
    <style type="text/css">
    body {
        margin: 0px;
    }
    #wrapper {
        width: 100%;
    }
    #juicebox-container-1 {
        float: left;
    }
    #juicebox-container-2 {
        float: right;
    }
    </style>
</head>
<body>
    <script src="gallery1/jbcore/juicebox.js"></script>
    <script>
    new juicebox({
        baseUrl: 'gallery1',
        containerId: 'juicebox-container-1',
        galleryWidth: '50%',
        galleryHeight: '100%',
    });
    </script>
    <script>
    new juicebox({
        baseUrl: 'gallery2',
        containerId: 'juicebox-container-2',
        galleryWidth: '50%',
        galleryHeight: '100%',
    });
    </script>
    <div id="wrapper">
        <div id="juicebox-container-1"></div>
        <div id="juicebox-container-2"></div>
    </div>
</body>
</html>

The problem appears to be with the location of your 'juicebox.js' file (or perhaps the entire 'jbcore' folder) on your web server.

According to the embedding code in your 'conference2' gallery, the corresponding 'juicebox.js' file should be located here:
http://www.ashworthphotography.co.uk/co … uicebox.js
... but going directly to that location in a browser results in an error 404 (file not found).

Likewise for your 'food' gallery.
According to the embedding code for this gallery, the 'juicebox.js' file should be located here:
http://www.markashworth.co.uk/portfolio … uicebox.js

Please double-check that you have uploaded the 'jbcore' folders to the correct locations on your web servers.

As you already have a 'jbcore' folder uploaded to the following location on your 'ashworthphotography.co.uk' domain:
http://www.ashworthphotography.co.uk/ga … al/jbcore/
.. you could point towards the 'juicebox.js' file in that 'jbcore' folder for your 'commercial2' gallery (as it is on the same domain).
For your 'commercial2' gallery, you could change:

<script src="jbcore/juicebox.js"></script>

... to:

<script src="/galleries/commercial/jbcore/juicebox.js"></script>

However, if you just upload a complete 'jbcore' folder to the following two directories, your galleries should display fine (without having to alter any embedding code).
(1) http://www.ashworthphotography.co.uk/conference2/
(2) http://www.markashworth.co.uk/portfolio/food/

If you already have 'jbcore' folders in these locations, then please check the permissions of the files and subfolders within them. Default permissions of 755 for folders and 644 for files should be fine.

As of v1.4.0, there is no longer a gallery preloader. This is noted in the Version History:

However, the image preloader remains and will be displayed (if showPreloader="TRUE") when an individual image is taking a while to load.

The image preloader seems to work OK in this Flickr-sourced demo gallery: http://juicebox.net/demos/lite/flickr/
Try viewing an image on a thumbnail page other than the first (where the images will not have been preloaded) and you should hopefully see the image preloader briefly before the image appears.

If you never see the image preloader in your own gallery, then please post the URL to your gallery's web page so that I can take a look and investigate further. Thank you.

It sounds like the 'New Gallery...' and 'Open Gallery...' buttons are non-responsive.
Is that correct? If so, try closing JuicenboxBuilder-Lite and manually delete the following folder from your hard drive.
Mac: /Users/your_username/Library/Application Support/Adobe/AIR/ELS/JuiceboxBuilder-Lite
Windows: C:\Users\your_username\AppData\Roaming\Adobe\AIR\ELS\JuiceboxBuilder-Lite

Now re-open JuiceboxBuilder-Lite and see if it works OK.

If this does not work, then try completely uninstalling and reinstalling JuiceboxBuilder-Lite following the procedure in this forum post.
(Just replace all instances of 'Juicebox-Pro' with 'Juicebox-Lite'.)

If you continue to experience difficulties, then perhaps you could upload a screenshot somewhere to show me what you are seeing and let me know what button or link you are trying to click (exactly what text is on the button or link) and what happens (or does not happen) when you click it.

Yes.

You can have a gallery with images that fill the page and without any thumbnails by using the following settings:

imageScaleMode="FILL"
showThumbsOnLoad="FALSE"
showSmallThumbsOnLoad="FALSE"
showThumbsButton="FALSE"
showSmallThumbsButton="FALSE"

You can have links in image titles and captions (and have multi-line image titles and captions) by using HTML formatting as noted in this FAQ:
How do I add HTML formatting to image captions and titles?

For example an image caption which contains two links on separate lines (an 'About' link and a 'Contact' link) would look something like this:

<caption><![CDATA[<a href="about.html">About Page</a><br><a href="contact.html">Contact Page</a>]]></caption>

I'm not sure what 'internal data' and 'control data' you are referring to.
Also, JuiceboxBuilder-Lite does not require or use Java at all.
It needs Adobe AIR which can be downloaded from here: https://get.adobe.com/air/
One you have Adobe AIR installed, you can following the steps in the Getting Started guide.
If you run into any problems installing JuiceboxBuilder-Lite, the Installation Issues section of the JuiceboxBuilder User Guide should hopefully help.

If you continue to experience difficulties, please let me know what you are trying to do and what error messages are displayed on screen. With a little more information, I should hopefully be able to help further. Thank you.

@wvdm

It sounds like you may not have completely removed your older version of JuiceboxBuilder-Pro.
As well as deleting the 'JuiceboxBuilder-Pro' file from the Applications folder and emptying your Trash, try searching your hard drive for the term 'JuiceboxBuilder-Pro' and delete any and all entries found (to be absolutely sure that nothing remains).

Also, when installing the latest version of JuiceboxBuilder-Pro, make sure that you do not have JuiceboxBuilder-Pro on any external drives attached to your computer. Unmount all drives other than main drive (including time machine backups) and retry installation.