Using relative links within your gallery's embedding code should work.
After making changes to your gallery, you may need to clear your browser's cache before reloading the gallery.

Currently, entering 'www.fabianbond.co.uk' into a browser's address bar redirects the user to 'www.fabianbond.co.uk/FabianBond/FBHome.html'. If you can redirect to 'fabianbond.co.uk/FabianBond/FBHome.html' instead, this should also work.

Different browsers likely have different ideas about what the best algorithm is for dynamic rescaling of images.
Your images do not look blurry in Firefox 15.0.1 on my PC. If anything, they look a little over-sharpened and jagged lines are visible on diagonals.
It might require a little trial and error in creating your original images (perhaps by increasing the quality and reducing the sharpening) before you find a setting which produces images which look acceptable in all browsers.

Make sure you embed each new gallery into a <div> with a unique 'id'.
After embedding the first two galleries, the embedding code for the next one would look like this:

<script src="http://www.example.com/gallery3/jbcore/juicebox.js"></script>
<script>
  new juicebox({
      containerId : 'juicebox-container3',
      baseUrl : 'http://www.example.com/gallery3/',
      galleryWidth: '100%',
      galleryHeight: '600'
  });
  </script>
<div id="juicebox-container3"></div>

If you reuse 'juicebox-container' ids, then on your home page (when many posts are displayed), there will be different galleries using the same 'juicebox-container' id and problems will occur.

You can embed multiple Juicebox galleries in a single post manually if you wish.

The process is as follows:
(1) Create your Juicebox galleries with the method of your choice, e.g. Lightroom plugin
(2) Upload the complete gallery folders (not just the contents) to your web server via FTP. It does not matter where you upload the folders to as long as you know where they are.
(3) Embed each gallery in your post using the baseUrl method as described here.

A few things to bear in mind:
(1) Make sure, when entering the embedding code into the body of the post, that the method of entry is 'HTML' rather than 'Visual'.
(2) You will need to embed each gallery into a <div> with a unique 'id'.
(3) In a WordPress environment, it might be easier to use absolute URLs rather than relative URLS in your embedding code in order to point to the gallery folders.
(4) When embedding multiple galleries in the same page, it is necessary to load the 'juicebox.js' file only once.

With this in mind, take the following as an example.
Say you have two gallery folders named 'gallery1' and 'gallery2' and you have uploaded them to the root of your web domain, in this example named 'www.example.com'.
Your embedding code would look something like this:

<script src="http://www.example.com/gallery1/jbcore/juicebox.js"></script>
<script>
  new juicebox({
      containerId : 'juicebox-container1',
      baseUrl : 'http://www.example.com/gallery1/',
      galleryWidth: '100%',
      galleryHeight: '600'
  });
  </script>
<div id="juicebox-container1"></div>
<script>
  new juicebox({
      containerId : 'juicebox-container2',
      baseUrl : 'http://www.example.com/gallery2/',
      galleryWidth: '100%',
      galleryHeight: '600'
  });
  </script>
<div id="juicebox-container2"></div>

Thank you for providing your login details. (I have left everything as I found it.)

Having run a few tests, I can confirm that as far as your server is concerned, your main page and your pop-up window are one and the same page and they cannot be distinguished using $_SERVER["SERVER_NAME"] . $_SERVER["REQUEST_URI"] as the same value is returned for each.
I have also tried $_SERVER["HTTP_REFERER"] and $_SERVER["PHP_SELF"] but to no avail.

As there is no way to check using PHP whether the shortcode_handler() function is being run inside a frame or pop-up window, the only other possible solution I can think of at the moment is to:
(1) Disable processing of the Juicebox shortcode by default.
(2) Pass a query string variable in the URL to be opened by your pop-up window.
(3) Check (within the shortcode_handler() function) to see if the variable is present and process the shortcode if it is.

This would require that you change your shortcode_handler() conditional to the following:

if (is_singular() || $_GET['jb'] == true) {

... and you would also have to ensure that your pop-up window links are changed from:

<a href="http://www.alexis.it/fashion-week/parigi/parigi-guy-laroche-en-bleu" rel="#pin-1622" class="whitebox">

... to:

<a href="http://www.alexis.it/fashion-week/parigi/parigi-guy-laroche-en-bleu?jb=true" rel="#pin-1622" class="whitebox">

If the code that generated the links currently look like this:

... then change it to:

<a href="<?php the_permalink(); ?>?jb=true" rel="#pin-<?php the_ID(); ?>

This solution should work fine.
I do not know where the code for your links is generated within your theme but I tested it on your main page and it works as expected.

With regard to the CSS clashes resulting in odd formatting of your galleries, currently the only solution (other than changing the CSS within your theme) would be to create your galleries on your computer (i.e. not using WP-Juicebox) and then embed them in iframes as I have previously suggested.

When I view your gallery, 15 images are displayed.
If I enter your gallery's Flickr configuration options into Flickr's own API Explorer, 15 images are returned.
(Go to the web page in the link above, enter your user_id (87772524@N06) and your chosen tags (travel), select the 'Send' checkboxes for these two items and click the 'Call Method...' button at the bottom of the page.)
If I remove the tags option from the search, all 17 images are returned.

The missing images are 'Francoi and his donkey' and 'French and American Co-pilgrims' but I do not see anything obvious about these two images that make them stand out from the others.

When tagging images in Flickr, it can take some time (up to 48 hrs) for the tags to propagate fully through Flickr's system in order for the images to show up in API searches so perhaps they will show up in time. Take a look at the Flickr: Help: Tags page for further suggestions.
If, after 48 hrs, the two missing images still do not show up, I would try re-uploading them to Flickr and re-tagging them.

A workaround would be to group your images within Flickr as a set and display the set using Juicebox's flickrSetId configuration option.

@paulr

This is due to the fact that the gallery is displayed dynamically using JavaScript and XML and the images are not simply displayed on the page using traditional <img> tags.
One way to achieve what you are looking to do would be to link directly to the images within your gallery's captions.
If creating or editing your gallery with JuiceboxBuilder-Pro, this can be done automatically by selecting 'Images -> Captions -> Use Link URL' from the drop-down menu at the top. The caption fields for all images will then be filled with links to the images so that users can right-click and save the image directly.

The latest code you posted is correct, replacing the if (is_singular()) { line in the shortcode_handler() method in the 'wp-juicebox.php' file. Try removing the trailing slash from the "www.alexis.it/" part of the code.
It may not work with all web servers but the solution to your problem lies with finding a conditional expression to replace if (is_singular()) { with such that the Juicebox shortcode will not be processed on your main overview page.

I would need access to your web server to test what is actually returned by using the different $_SERVER elements so trial and error may be required.

To remove the drop-shadow from all elements within your gallery, open the file 'jbcore/classic/theme.css' and search for and delete all lines which contain the term 'shadow'.

Captions can be positioned horizontally via the captionHAlign Pro Option, whose value can be LEFT, CENTER or RIGHT.
If you continue to experience difficulties, please post the URL to your gallery so that I can take a look.

WordPress will either parse the Juicebox shortcode and display the gallery or not. In a traditional WordPress theme (with no pop-ups), my code above will display the gallery when the post is viewed on its own page but will not display the gallery on the main overview page.
WordPress does not seem to classify your pop-up as being the post's own page (as the is_singular() method returns 'false' when the post is viewed in your pop-up).
Rather than try to determine whether or not the code is being parsed within your pop-up (and display the gallery if it is), you could instead try to prevent the Juicebox shortcode from being processed on your main overview page by changing the following code in the shortcode_handler() method (from the 'wp-juicebox.php' file) that I posted above.
Change:

if (is_singular()) {

to:

$pageURL = $_SERVER["SERVER_NAME"] . $_SERVER["REQUEST_URI"];
if ($pageURL != "www.alexis.it/") {

@TexStarPhoto

Try using the following settings in your gallery's XML file:

imageScaleMode="SCALE"
imagePadding="0"
stagPadding="0"
showLargeThumbs="FALSE"

A list of all Juicebox-Pro configuration options can be found here.

Yes, it should (in a standard WordPress page). Currently this is not the case with WP-Juicebox where the user must also either 'Edit' and 'Update' the post containing the gallery or 'Edit' and 'Save' the gallery itself to rebuild the gallery's XML file with the new information.
However, this behavior will change in the next verion of WP-Juicebox and the user will no longer have to edit the post or gallery in order for changes in the Media Library to take effect in the corresponding gallery.