So, is there any way to achieve what I asked for in the first post. I.e. on the iPhone it would jump directly into SSM without any splashing and directly show the thumbnails, without two additional clicks, or more precisely taps, on the touchscreen.
Gallery #3 seems to do this (going straight to the thumbnail page without a Splash Page).
gall03 works like Gall01 on PC (LSM), but on iPhone it locks into LSM,
On my iPod Touch, Gallery #3 displays in Small Screen Mode (not Large Screen Mode). Only Small Screen Mode displays the thumbnails and main images on different pages. (Large Screen Mode would display the thumbnails and images on the same page.)
and everything is way too small.
Try scaling your web page for mobile devices by using the following <meta> 'viewport' tag in the <head> section of your page (rather than the one you currently use):
<meta name="viewport" content="width=device-width, initial-scale=1">
The 6 Plus has a Full HD screen, way bigger than the SSM 1024x768 threshold, but Juicebox went into SSM.
Juicebox takes into account the device itself as well as the screen size when deciding which Screen Mode to use. It is not possible to change the internal logic which determines the Screen Mode but you can force a certain Screen Mode to be used via the screenMode configuration option (although I understand that you do not want to do this).
What also happened, was that there was no way to hide Safari's own top and bottom bars
Apple removed 'minimal-ui' from iOS 8 (which was used to remove the bars from the screen in previous versions of iOS).
Your web page still includes 'minimal-ui' (in the <meta> 'viewport' tag) which is why the bars still disappear on devices running versions of IOS prior to 8 (but not on the iPhone 6 Plus which I expect is running iOS 8).
Is it necessary to create the small images at all? My iPhone has a 960x640 display, and does not use the small pics for anything.
This is entirely up to the author of the gallery to decide and might be somewhat dependent on the target audience for your web site. For example, if it is highly likely that visitors to your web site will be viewing your web pages on large desktop monitors (rather than small mobile devices), then you could perhaps get away with not having to provide small images in your galleries.
Incidentally, as you switch between galleries using links in a header, you might like to take a look at the Using a Resizable Gallery with a Header support section.
The View Resizable Gallery with Top Menu Example uses a header with links to different galleries (like your web pages do) and the remainder of the browser window space is used for the galleries without the need for any vertical scroll bars (which your web pages currently use).
You could view the source of the demo web pages in a browser and copy/modify them to suit your own needs.