This is certainly the case on iOS devices (and is documented in the description for playAudioOnLoad on the Config Options page) but I do not know if this is also true for Android devices (and you should not need to click the Audio Button twice). It sounds like your gallery's Audio Button may initially be displaying the 'pause' symbol (implying that the audio is playing when it is not) but then you have to click the button (to display the 'play' symbol) and then click the button once again. Is this correct?

Please make sure that you are using the latest version of Juicebox-Pro (v1.3.3) to ensure that any bugs that were present in previous versions but which have since been fixed are not contributing to your problem.
Instructions for upgrading Juicebox-Pro can be found here.

If you continue to experience difficulties, please let me know what device/browser you see this problem in and please post the URL to your gallery so that we can investigate further. Thank you.

The Dropbox link you quoted is, indeed, a download link. It offers the 'index.html' file to the user as a download rather than allowing the user's browser to display its contents like a regular web page.

Have you tried creating a 'Public' folder by logging into your Dropbox account and clicking this link?
https://www.dropbox.com/enable_public_folder

It looks like the <iframe> code itself has been inserted into your Squarespace page correctly. (The problem is just getting a Dropbox public link to your gallery's 'index.html' file.)
If you want to check that the <iframe> code is OK, just replace the Dropbox download link (with is currently used as the 'src' attribute of the <iframe>) with any other web page (such as http://www.google.com) and you should see the web page you choose displayed within your own web page.

If you are using the Dropbox desktop application, then navigate inside your gallery folder within your Dropbox 'Public' folder, right-click the 'index.html' file and select 'Copy public link' from the right-click menu options.
If you are using the Dropbox web brower interface, then navigate inside your gallery folder within your Dropbox 'Public' folder and click the 'Copy public link' button on the toolbar at the top (as in the first screenshot on this web page).
If you do not see such an option, then it is possible that your Dropbox account does not yet have a 'Public' folder.
If this is the case, then you can create one by following the 'Creating a Public folder' instructions on the same page (link above).

There certainly seems to be no problem with the gallery files themselves. The issue does, indeed, seem to be 3G related.

I was not sure what you meant by "I have a Juice block.htaccess file".

The file should be named .htaccess (ie. just a file extension - no actual name) and should be located in your web space's root directory. The file should contain the following code.

<IfModule mod_headers.c>
Header set Cache-Control "no-transform"
</IfModule>

This depends entirely on the structure of the Juicebox galleries on your static site.
If each gallery is a self-contained entity in its own folder, then this will simplify matters and all you need to do is copy across all the complete gallery folders.
However, if your galleries currently share a 'jbcore' folder or all of your galleries draw images from a single directory of images (or the gallery structure is somehow non-standard), then it may be more difficult to separate out the gallery files from all the other files on your web server.
As long as all your Juicebox galleries are clearly separate from each other, then there should not be a problem.

Whether you do this now or later is up to you. As long as you use relative paths in your embedding code and gallery XML files, then you should not need to change any paths when changing domains.

It might be wise to try the process with one of your galleries just now to see if you encounter any unforeseen problems which could perhaps be worked out ahead of time.

After clicking a Splash Page to expand a gallery, a user would ordinarily click the 'Expand/Close' button on the gallery's Button Bar to close the gallery and return to the original page containing the Splash Page.

In Small Screen Mode (used by default to display the gallery on mobile devices), the Button Bar is always displayed on the overlay (as screen real estate is limited and to reserve space for a 'top' area on a small screen device would not be ideal) and your gallery sets showImageOverlay="NEVER", so the 'Expand/Close' button is not visible.

Possible solutions:
(1) Set showImageOverlay="AUTO" or "ALWAYS"
(2) Set showInfoButton="TRUE" which overrides the default behavior of the overlay and always displays the Button Bar (with the Info Button to toggle the overlay on and off)
(3) Set screenMode="LARGE" so that Juicebox uses Large Screen Mode (with the 'top' area) to display the gallery on all devices and in all browsers
(4) You could also use the Back Button to link directly to your web page (but this also relies on the overlay being visible in Small Screen Mode):
showSmallBackButton="TRUE"
backButtonURL="http://www.example.com"

If you set showImageOverlay="AUTO", you will see the icons that you refer to (the Button Bar) and you will have access to the 'Thumbnail' button (to switch between the thumbnail and image pages) and the 'Expand/Close' button to allow the user to return to your main page.

I can successfully view your http://pictureslondon.co.uk/Lizy%20Brun … index.html gallery on an iPod Touch 4 (running iOS 6.1.5) so I would expect that it should also display fine on an iPhone 4S.
Make sure that your iPhone 4S is using wi-fi (and not 3G) and try completely clearing your browser's cache before reloading your web page.
If you are using a 3G connection, then please see this FAQ:
Why can't I view my gallery on a 3G mobile connection?

I have been able to replicate the problem you describe and it seems to be directly related to the use of the Splash Page.

When a Juicebox gallery is displayed on a mobile device, Small Screen Mode is used and, by default, the Splash Page is displayed. This may not happen on an iPad with a larger screen (where Large Screen Mode may be used) which is perhaps why you do not see the problem on your iPad.
(For more information about Screen Modes, please see here.)

If I set showSplashPage="NEVER" (to prevent the Splash Page from being displayed) or screenMode="LARGE" (to display the Large Screen Mode version of the gallery which, by default, does not display the Splash Page), the infinite scroll page does not jump back up to the top.

I have contacted the developers regarding this issue and hopefully it can be fixed in a future version of Juicebox but in the meantime, a workaround would be to prevent the Splash Page from being displayed (by setting either showSplashPage="NEVER" or screenMode="LARGE").
Please note that both of these configuration options are available in Juicebox-Pro only and are not supported in Juicebox-Lite (the free version).

The only oddity I see on your web page at the moment is that your embedding code (for your November gallery) is enclosed in CDATA tags and that the closing tag has its final > character escaped as &gt; in the following line:

// ]]&gt;</script></p>

It may (or may not) make a difference but try either replacing the line above with:

// ]]>
</script>
</p>

... or completely remove the CDATA tags around the JavaScript code.

Is it OK in both Chrome and Mobile Safari on the iPad? What iOS does your iPad run? Perhaps you can upgrade the version of iOS on your iPhone (if it is not already up-to-date) to see if this makes a difference. (By checking both Chrome and Mobile Safari in both an iPhone and an iPad and checking which version of iOS each device is running, you may at least be able to find a pattern within iOS devices.)

As you seem to be embedding your galleries manually (rather than using WP-Juicebox), then perhaps a workaround would be to upload your gallery folders to your web server and include links to them within your posts (rather than embed them directly in your posts).

My suggestion was not to replace the line:

winWidth = window.innerWidth ? window.innerWidth : $(window).width();

... but to make the adjustment after it, e.g.:

winWidth = window.innerWidth ? window.innerWidth : $(window).width();
winWidth = winWidth - 20;

The code which determines the window width will return the result in pixels so you should stick to using a pixel value for this right-hand border so you can subtract the border (px) from the window width (also px).
Also, when making the adjustment to the winWidth value, the code will be purely arithmetic so you should not use a suffix, such as px. (I used a suffix in my original code above but I have corrected the error. Sorry for any inconvenience.)
Use a CSS border such as:

body {
    margin-right: 20px;
}

... and then use either:

winWidth = window.innerWidth ? window.innerWidth : $(window).width();
winWidth = winWidth - 20;

... or:

winWidth = window.innerWidth ? window.innerWidth - 20 : $(window).width() - 20;

I have tested this myself and it seems to work fine.

The Showkase Acqua Photo demo uses Juicebox-Pro. (The Juicebox galleries within the demo use Pro options such as AutoPlay and hiding thumbnails.)

To achieve this, you would need to either:
(1) Purchase Showkase-Pro (which comes with Juicebox-Pro)
... or:
(2) Purchase Showkase-Standard and Juicebox-Pro separately

You can change the rollover tooltip text using the languageList configuration option. Please see here for details.
Change the default 'Go Back' text to whatever you like and add the languageList configuration option as an attribute to the opening <juiceboxgallery> tag in your gallery's XML file.
For example:

<juiceboxgallery
    galleryTitle="Juicebox-Pro Gallery"
    languageList="Show Thumbnails|Hide Thumbnails|Expand Gallery|Close Gallery|Open Image in New Window|Next Image|Previous Image|Play Audio|Pause Audio|Show Information|Hide Information|Start AutoPlay|Stop AutoPlay|AutoPlay ON|AutoPlay OFF|Email|Buy this Image|Share on Facebook|Share on Twitter|Share on Google+|Share on Pinterest|Share on Tumblr|of"
>

The backButtonText configuration option is only for use when using text for the Back Button instead of an icon (not for the rollover text which is handled by the languageList configuration option).

Please post the URL to your gallery so that I can take a look for myself and help further.