There is a lot going on within your web page and as it loads, you can visibly see elements shifting around until they settle in place (as the JavaScript and CSS is being rendered by the browser).

There are differences in your web page, not just between versions of Internet Explorer but also between different browsers.
For example, in Firefox 19.0.2, your 'Select Language' combobox is initially completely obscured by the gallery (which it is not in IE10, Chrome 25 or Safari 5.1.7) but it partially appears if you resize the browser window and then maximize it.

As you are expressing your galleryWidth and galleryHeight as percentages, check all parent containers of the Juicebox container (your 'box_slideshow' <div>) to ensure that they all have heights specified via CSS and please see the Using Percentage Heights note in the embedding guide. This suggests using inline CSS on the parent containers rather than on the Juicebox container (your 'box_slideshow' <div>) itself.

If there is a problem with the gallery not being able to determine it's height in certain browsers, then you could perhaps express your gallery's height as a fixed pixel value rather than as a percentage.

Another workaround might be to isolate the gallery completely from the rest of your web page by embedding it using an <iframe>, documented as Option #2 here. As your gallery does not use the Expand Button or Back Button, there should be no drawbacks to using an <iframe>. This should avoid any conflicts with your main page and should hopefully work in all browsers.

The link you have provided redirects to http://afrika.delosgaia.nl/index.htm and there seems to be only a SimpleViewer gallery on the page. I see neither a Juicebox-Pro gallery, nor the code you posted (only <div id="box_slideshow"></div>).

I'm not sure it will help (as I have not yet seen the page containing your Juicebox-Pro gallery) but take a look at this FAQ (it relates to IE9 but may also be relevant to IE8):
My gallery looks strange in Internet Explorer 9. Why?

Also, if you are embedding your Juicebox-Pro gallery directly into your 'box_slideshow' <div>, then try removing the 'style' attribute from the <div>, set your gallery's dimensions using the galleryWidth and galleryHeight configuration options (in the embedding code) and, if expressing galleryWidth and galleryHeight as percentages, check all parent containers of your 'box_slideshow' <div> to ensure that they all have heights specified via CSS.

There have been a couple of issues specific to IE8 (regarding the opacity of thumbnail frames and the the FADE image transition type) but these have already been addressed and will be fixed in the next version of Juicebox-Pro.
Just to let you know, there are no known issues regarding gallery sizing in IE8.

If you continue to experience difficulties, please post the link to your Juicebox-Pro gallery so that I can take a look and investigate further.

This is not possible.
The only way to display the grid of thumbnails is to set screenMode="SMALL" but the Large Screen Mode configuration options will work only in Large Screen Mode. There is no way to mix or change the screen modes in a gallery.

A Splash Page is a placeholder for the gallery: an image which, when clicked, displays the gallery itself.
This is different from the thumbnail grid displayed in Small Screen Mode.
If you always want the thumbnail grid to be displayed, set screenMode="SMALL".
If you always want the Splash Page to be displayed, set showSplashPage="ALWAYS".

In order to display a Juicebox gallery, your web page must load the 'juicebox.js' file (which, in turn, loads the 'theme.css' file and other related resources).
However, there is no way to speed this up (other than to perhaps host your images locally rather than have Juicebox fetch them from Picasa's servers). WP-Juicebox embeds a Juicebox gallery using the same method as is described in the Embedding Guide with the exception that your WordPress pages and the Juicebox XML file are created dynamically by PHP when the page and gallery are viewed. If all pages were static, things might run a little quicker (as your server would not need to create the pages on demand) but this is not how WordPress works. The advantage of having the Juicebox XML file created dynamically is that you can change your gallery's images and captions directly in Picasa without having to manually rebuild your Juicebox XML file afterwards.

On this web page: http://www.irmucha.pl/nieswiateczne.html
... it looks like the initial gallery is being displayed by the following code:

$(document).ready(function () {
    doLayout();
    $(window).bind('resize', doLayout);
    new juicebox({
        containerid : 'juicebox-container'
    });
});

... and this instance of Juicebox does not have a valid 'config.xml' file in the correct location.
Try adding the baseURL configuration option (from the embedding code further down your page) to the code above, i.e.:

$(document).ready(function () {
    doLayout();
    $(window).bind('resize', doLayout);
    new juicebox({
        containerid : 'juicebox-container',
        baseUrl : 'nieswiateczne/'
    });
});

I expect that you may not need both instances of embedding code on your page.
Once you have added the baseURL configuration option to the first instance (at the top of the page), try removing the second instance (at the bottom of the page).

With the single-site license, you can have an unlimited number of galleries on an unlimited number of pages (as many galleries per page as you like) as long as all your galleries are hosted on the same domain (and any subdomains).

There is no App (or any other method) that will allow you to create and upload a Juicebox gallery from a mobile device.
As an alternative to my Flickr suggestion, you could first set up a Juicebox gallery (on your computer) using a PHP file to read the contents of a directory and to display the images contained with that directory in your Juicebox gallery.
Please see this forum post for an example of such a PHP file.
You could then upload your images to the specified directory on your own web server using an FTP App.

Thank you for providing the URL to your gallery.
You have chosen to display the 'Info Button' in your gallery.
As noted in the description for the showInfoButton configuration option in the Button Bar Options section of the Config Options page:

You can now toggle the overlay on and off by tapping on the Info Button rather than anywhere on the screen.

@marcrmiller

Your gallery displays and functions OK in IE10 on my PC (and also in Firefox 19.0.2, Chrome 25 and Safari 5.1.7).

Please note that as you have use the www.cal500.com subdomain in the URLs within your embedding code, your gallery will display only when you access your web site using the www.cal500.com subdomain (and not when going to the main cal500.com domain).
All gallery files must be on the same domain (or subdomain) as the web page containing the embedding code due to the JavaScript same origin policy.

One solution to this would be to use relative URLs (rather than absolute URLs) in your embedding code.

Juicebox does not currently have the ability to horizontally or vertically align the main images within the image area.
We plan to introduce new configuration options to allow such positioning in the next version of Juicebox-Pro (though I do not know when it will be released).

@contactgeoff

I notice that your gallery uses Juicebox-Pro v1.1.1. Please try upgrading to the current version (v1.2.0) to see if this makes a difference.
Please see the Upgrading Juicebox page for details.