What I can see is that the option does not allow us to open a new tab but the gallery will stay on the same page in some way.
When using expandInNewPage="TRUE", Juicebox will open the expanded gallery using a new page (the 'full.html' page within the 'jbcore' folder) but within the same tab/window. It is not possible to expand the gallery in a different tab/window.
It looks like something on your web page (most likely some JavaScript) is preventing WebKit browsers (Chrome, Safari and Opera) from opening the gallery using the Fullscreen API, even though your gallery uses useFullscreenExpand="TRUE".
Unfortunately, your web page loads over 20 external JavaScript files and also contains several internal <script> sections so troubleshooting the page may be quite difficult and some trial and error may be required to find the cause of the problem.
I would start by deactivating all plugins to see if the gallery expands as expected and then reactivate your plugins one-by-one (checking the gallery each time) to see if you can find a conflicting plugin. (The problem may or may not be related to the Jetpack plugin.)
I still see a weird display of my September gallery post (main image is very little, with no thumbnails) after I opened November gallery first (from the corresponding post).
I have expanded your September gallery (after expanding and closing your November gallery) in Chrome 32 on my PC but I do not see the issue you describe (though the gallery does not use the Fullscreen API, as I noted above). Perhaps the problem is isolated to WebKit browsers on the Mac.
Try clearing your browser's cache before reloading your web page to ensure that your browser is fetching and using the most recent gallery files from your web server. If your galleries are set up identically (double-check the settings for each gallery), then I would expect them to display and function identically. If your browser is caching and using older gallery files, then this might explain the problem you are reporting.
Your gallery uses Social Sharing (Facebook) and the # is required to give each image in your gallery a unique URL.
On iPad, after closing a gallery, I am often in a situation in which the infinite scroll won't work anymore within the page...
Again, this sounds like there may be some conflicting JavaScript on your web page causing this problem. (Although, this time, it sounds like it may be a direct conflict between Juicebox and the Jetpack plugin.)
Deactivating and reactivating your plugins might help locate the source of the problem.
Is there any option to open the gallery in a new tab?
A gallery cannot be expanded in a new tab/window but a possible solution would be to simply include links to your galleries in your WordPress posts. You could essentially create your own Splash Pages which would consist of an image (or just text) which link to 100% x 100% galleries in new tabs/windows by using the target attribute of the HTML <a> tag.
A text link to a gallery would look something like this:
<a href="/gallery/index.html" target="_blank">Click here to open gallery in new tab/window.</a>
You could then use the Back Button in your galleries to allow users to return to the original web page.