Thank you for keeping me updated.
I'm glad I was able to at least point you in the right direction.
I hope you can find a combination of settings that gives you the balance and functionality that you are looking for.

I have written a PHP script to select a certain number of images from a pool of many and display them in a random order but I have not been able to replicate the problem you describe.
The images displayed (and their order) change each time the gallery is displayed (just like your gallery) but the images retain their order when the gallery is expanded.
(I have also tried setting expandInNewPage to AUTO, TRUE and FALSE with the same results each time.)

From your own scenario, it certainly looks like Juicebox refers to the XML file (seeing a new set of images) when the gallery is expanded.

I think the problem may be due to web server cache settings.
If your browser has cached the XML file, then Juicebox should be able to refer to the cached version and display the same images in the expanded gallery as in the normal version.

However, if your web server does not allow browsers to cache files and forces browsers to fetch files from the server each time they are required, then Juicebox would have no option but to use a freshly-generated XML file with a new set of images when the gallery is expanded.

Please check your web server's Cache-Control headers. Allowing browsers to cache files may solve your problem.
Try adding the following code to an .htaccess file in your root directory (as long as you have an Apache web server).

<IfModule mod_headers.c>
    Header set Cache-Control "max-age=86400"
</IfModule>

Alternatively, try setting the header via PHP (at the top of your 'config.php' file) with the following code:

header("Cache-Control: max-age=86400");

I hope this helps.

Clearing your browser's cache should work. If your browser has no cached version of your gallery's XML file, then it has no option but to use the current version from your web server.

Please see this FAQ:
When I update my gallery I don't see my changes online. Why?

It sounds like your gallery's XML file might not have been updated on your web server. If you are in any doubt that your old file is not being overwritten with the new version, then delete the old version from your web server first before uploading the new version in its place.

If you want to prevent browsers from caching your gallery's XML file, then try defining your gallery's XML file with a unique identifier (e.g. the current time which will be different each time the browser loads the page) using the configUrl option in your gallery's embedding code.

<!--START JUICEBOX EMBED-->
<script src="jbcore/juicebox.js"></script>
<script>
    new juicebox({
        containerId : 'juicebox-container',
        configUrl : 'config.xml?nocache=' + new Date().getTime()
    });
</script>
<div id="juicebox-container"></div>
<!--END JUICEBOX EMBED-->

If Readymag does not support FTP (please check with their support), then there will be no easy way for you to upload your gallery files to the web space that they provide.
You would need to host your gallery elsewhere (such as Google Drive) and load your gallery into an iframe on your Readymag web page.
Please see the Embedding in a Web Template Site instructions.

When following the web template site instructions, you can create your gallery (and configure it) using JuiceboxBuilder-Pro before uploading the complete gallery folder (all images and gallery resource files) to Google Drive.

Unless you are using a Flickr account as a source of images, the images to be displayed in the gallery need to be listed in the gallery's XML file. When creating a gallery with JuiceboxBuilder-Pro, the images are included within the gallery folder and are usually hosted on the same domain along with the gallery resource files and the web page containing the gallery's embedding code. You could host your images elsewhere but you would need to manually edit the gallery's XML file and ensure that the imageURL entries point towards them.

Juicebox galleries adapt to different devices and browser window sizes so there should be no need to create separate galleries for individual scenarios.
Please see the Gallery Tour for details of Screen Modes and Input Modes.
Also, you might like to look into the possibility of creating a Multi-Size Image gallery so that images of an appropriate size are using in the gallery depending on the device and browser window size being used to view the gallery.

Yes. As long as all your galleries are on the same domain, there there should be no problem with them sharing a single 'jbcore' folder somewhere on your server. (It does not matter whether the galleries are embedded in existing web pages alongside other content or embedded on pages of their own.)
For each of your galleries, just check the 'index.html' file for the following line of code and adjust the path so that it points to your shared 'jbcore' folder.

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

I'm afraid that we have no payment options other than Gumroad and PayPal and if neither Gumroad nor PayPal will allow you to make purchase with a credit card in your country, then there is unfortunately little that we can do.
Perhaps if you have a friend who has a PayPal account or lives in a different country, then they could purchase Juicebox-Pro for you and you could pay them back.

Please let me know what generation of iPad you have and what version of iOS you are using.
If possible, try upgrading to a more recent version of iOS.

Also, try a web search with terms such as "iframe iPad Mobile Safari". I do not know if any of the problems that crop up are directly related to your own problem but there are certainly other users who seem to be experiencing iframes which are not behaving as expected in Mobile Safari on iPads. One of the top search results might help or at least point you in the right direction.

Another thing that might help is rather than setting a percentage height for your iframe, you could perhaps use JavaScript to determine the actual required height of the iframe in pixels (depending on the size of the user's browser window) and then set the iframe's height to this pixel value dynamically using JavaScript and CSS.

That certainly sounds strange. The height of your iframe should be consistent on your web page (and determined either via CSS or by the iframe's 'height' attribute) no matter what content is loaded into the iframe.
Ordinarily, the content of the iframe should not change the layout of the parent page unless there is specific JavaScript code designed to do so and Juicebox does not contain any such code.

Your gallery looks to be a standard 100% x 100% gallery so it should fill the iframe and there is little you can do on the gallery side to change things.

From your screenshot, it looks like your gallery is filling the iframe (and everything seems to be correctly vertically centered on the page) but the iframe is not tall enough for the complete content of the page to fill your iPod browser window.
The problem seems to be with the height of the iframe on your main page.

I do not know how the height of your iframe is calculated (your web site is rather complicated).
Maybe it is looking at the content of the iframe and trying to determine the height required. Perhaps a gallery with 100% height is somehow difficult for your JavaScript code to figure out how tall the iframe should be.
Check the code that sets your iframe's height to see if you can spot any potential problems with it.

Also, please check to see if you see the same problem in Mobile Chrome that you see in Mobile Safari.
Thank you.

If possible, please upload a screenshot so that I can see what you are seeing on your iPod.
The layout might be similar to what I see when I view your gallery in a desktop browser (the gallery being displayed to the right-hand side of the browser window).
The iframe your gallery is loaded into is nested within several parent containers and you may need to check the CSS of these divs.
(It looks like your web page may be designed for a specific device or browser width.)

To see if the problem is directly related to Juicebox or not, try loading a different page into your iframe.
Try loading something like http://www.google.com into your iframe rather than your gallery's 'index.html' page to see if the same layout problem happens on you iPod.

That's to be expected (on Google Drive or any other host). The 'full.html' file is for Juicebox's own internal use only (when a gallery is expanded in a new page of its own) and may not display the gallery when accessed directly (depending on the gallery's structure).

However, trying to access the 'full.html' page directly might actually be quite a good test.
This essentially takes Juicebox out of the equation. All we are trying to do is access an HTML file directly.
The HTML page should still be rendered by the browser (displaying the 'Config XML file not found.' message).

In my own gallery, going directly to the 'full.html' page in Mobile Safari on an iOS device:
https://7e149673cba79128b4d3fe7ac7a9b89 … /full.html
... displays the page (with the 'Config XML file not found.' message) as expected.

However, going directly to the 'full.html' page in your gallery in Mobile Safari on an iOS device:
https://3fd620ce7dd6a53cea4c9215317f4dd … /full.html
... displays the error 400.

Here are a few things to try.
(1) Change the content of the 'full.html' file (temporarily) to something like the following which has nothing at all to do with Juicebox and try to access the file directly in Mobile Safari to see if the same problem happens.

<!DOCTYPE html>
<html lang="en">
<head>
    <title>Test</title>
</head>
<body>
    <div id="test">Test text.</div>
</body>
</html>

If this still does not display correctly in Mobile Safari when accessed directly, then there may be an issue with the privacy settings on your 'full.html' file. Try making it private and then public again.

(2) Upload a completely new HTML file (with the sample code above) to the same directory as the 'full.html' file (your 'jbcore' folder) and see if you can access this directly in Mobile Safari. This should determine whether or not the problem is specifically with the 'full.html' file or with all HTML files in that directory.

(3) Try re-uploading your entire project to a new folder on Google Drive and ensure that all files and folders are made public.

The problem seems to be with access to the 'full.html' file on your Google Drive web space (both direct access and when Juicebox tries to access it dynamically) on iOS devices (but not in desktop browsers) so when I suggest that you try to access test pages directly (suggestions #1 and #2 above), you might want to use a service such as tiny.url to shorten the URLs and make life easier when entering the URLs into your Mobile Safari browser's address bar: http://tinyurl.com/

If we can figure out why the 'full.html' file cannot be accessed in Mobile Safari, then the other error 400 (which occurs when selecting a gallery from your menu) is likely to be related and hopefully the solution to one problem will fix the other.

Incidentally, I notice that your gallery's 'full.html' URL is much longer than mine. Maybe this has some relevance (especially when the 'juicebox.js' file is loaded and its code is run). Suggestions #1 and #2 above should tell us more.
If the problem does turn out to be due to the length of the URL, then there is little that can be done other than host your website somewhere other than Google Drive where you hvae more control over the path to your site.
If you have access to a regular web server, try hosting your web site there to see if everything works as expected. This should at least let us know whether or not the problem is somehow directly related to Google Drive (maybe with URL length or sharing settings).

Try reinstalling the plugin from scratch to ensure that all the required files are present and correct.
If you need to, you can download a fresh copy of the Juicebox-Pro zip file using the link from your purchase email.
I recommend that you first delete the existing 'juicebox_pro.lrwebengine' folder from your 'Lightroom\Web Galleries\' directory before reinstalling the plugin.

Also, if you are using Lightroom 6/CC 2015 but have not upgraded it recently, then you might like to try upgrading it to the latest version (6.3/CC 2015.3) which Adobe released a couple of days ago.
This might help.

If you continue to experience difficulties, please let me know what (if anything) you see in the live preview window.
Do you see just a blank space or an error message (or something else)?
Does the problem seem to happen with only the Juicebox-Pro plugin or with other Web Layouts, too?

Try installing the Juicebox-Lite plugin (from the Juicebox-Lite download zip file) to see if the same thing happens with this version.
You can install both the Juicebox-Lite and Juicebox-Pro plugins side-by-side at the same time.
Juicebox-Lite can be downloaded from the download page here.