With Juicebox-Pro, you could certainly produce individual galleries to each display a different Flickr Set.

With Juicebox-Pro, you could create as many individual galleries as you like but you would need to link them together manually using techniques such as those in the Embedding Multiple Galleries support section. Juicebox-Pro does not come with any automated method of linking together multiple galleries.

An alternative solution would be to use another one of our products: Showkase.
Showkase is a PHP web application which would allow you to create a complete portfolio web site online using a web browser interface.
Showkase has full support for Juicebox-Pro and your Flickr-based galleries could be created within the application itself (or created with JuiceboxBuilder-Pro and then imported into Showkase).
You could create gallery index pages and have as many galleries listed on each index page as you wish.
Each gallery is represented by a thumbnail image with the gallery title displayed below and the gallery is opened when the user clicks on the image.
(Showkase also has the ability to create non-gallery pages such as 'About', 'Contact' and 'Basic' pages where you can enter information about yourself and contact details.)
All of this is done automatically within the interface without the need for any manual coding at all.
If you would like to take a look around the Showkase interface, you can log into the Live Demo Admin here.
Demo sites can be found here and a sample Gallery Index page can be found here.
Showkase can be purchased as Showkase-Standard (which comes with Juicebox-Lite and SimpleViewer-Standard, the free versions) or Showkase-Pro (which comes with Juicebox-Pro and SimpleViewer-Pro). The only difference between Showkase-Standard and Showkase-Pro is the bundled viewers.

Flickr sets are always returned in the order specified on the Flickr set page.
For more details on how to change the order of images in a Flickr Set, please see this FAQ:
How do I change the order of images in a Flickr Photo Set?

You would need to add a new Gallery Page within your Showkase site each time you create a new Flickr Set. (There is no way for Juicebox-Pro or Showkase to know that you have created a new Flickr set.) However, this is a quick and easy process which can be done in a matter of minutes (and the new gallery can automatically be listed on a Gallery Index Page if you like).

Juicebox-Pro supports displaying the Flickr Title and/or Flickr Description in the gallery's caption area.
Please see the Flickr Options section of the Config Options page for details.
The caption area can then be positioned using the captionPosition configuration option.

Both Juicebox-Pro and Showkase are mobile-friendly.
Please try viewing the demo Juicebox galleries and Showkase sites in any mobile devices you have.
Juicebox: http://www.juicebox.net/demos/
Showkase: http://www.showkase.net/demos/

We do not have trial versions of Juicebox-Pro or Showkase but we do offer a money back guarantee:
Juicebox: Can I try a trial version of Juicebox-Pro?
Showkase: Can I try a trial version of Showkase?

I hope you find the information (and links) above helpful.

In order for a Juicebox-Pro gallery to display a Flickr Set, you would define the Set Id using the flickrSetId configuration option in the gallery's embedding code.
For example:

<!--START JUICEBOX EMBED-->
<script src="jbcore/juicebox.js"></script>
<script>
    new juicebox({
        containerId: "juicebox-container",
        galleryWidth: "100%",
        galleryHeight: "100%",
        backgroundColor: "#222222",
        useFlickr: "TRUE",
        flickrSetId: "72157645547079035"
    });
</script>
<div id="juicebox-container"></div>
<!--END JUICEBOX EMBED-->

With Showkase-Pro, you would be able to create a Juicebox-Pro gallery to display a Flickr Set by entering the Set Id into the Showkase-Pro interface. However, there is no way for Showkase-Pro to detect when a new Flickr Set has been created and no way for Showkase-Pro to automatically create a new Juicebox gallery to display the Set.
As I noted earlier, each time you create a new Set, you would have to create a new Juicebox gallery manually within Showkase. This is a quick and easy process (no coding is required at all) and might be a compromise you are willing to make in order to have a mobile-friendly site.