Several questions

The Gallery Index Page should use the entire width of the page (although there may be a small gap to the right of the page if there is not enough room for an extra thumbnail).
Please post the link to your Gallery Index Page and provide a screenshot so that I can see what you are seeing and I might be able to help further.
Also, if you are not already using the latest version of Showkase (v1.7.2), then please try upgrading to see if this helps.
For reference, full instructions for downloading the latest version and upgrading can be found here.

Unfortunately, Showkase was not set up for this and there is no easy way to display custom text below the page name. All the information displayed on the Gallery Index Page is taken from the 'pagesdata.txt' file. There is no gallery specific information stored in the 'pagesdata.txt' file and, if you wanted to display the Gallery Description (for example), then you'd need to write a PHP script to parse the corresponding gallery's 'config.xml' file to extract the information.

You can certainly rename a page on the 'Pages' tab (click 'Rename' between 'Edit' and 'View') but the corresponding folder (inside which all the page's files are stored) will always be named after the page name (this behavior cannot be changed) and it is the page name that is displayed below the thumbnail on the Gallery Index Page. You can give a gallery a Gallery Title (in the 'Lite Viewer Options' section) and the Gallery Title will appear in the gallery itself (as long as galleryTitlePosition is not set to NONE).

No. It is not possible to change the structure of a Showkase site. All page folders must remain in the root Showkase directory where Showkase created them.

You are certainly free to change the 'theme.css' file but Juicebox will not be aware of any changes that you make and might expect certain gallery elements to be in certain positions. Custom CSS modifications can often result in unexpected behavior (for example, when a browser window is resized and the gallery needs to be redrawn). If everything seems to be OK, then that is great. However, if you find that this is causing problems, then I would recommend using the available configuration options (such as captionPosition) instead.

If thumbsPosition is set to TOP or BOTTOM, then maxThumbRows is used as an absolute value (and the number of thumbnails rows will not change).
If thumbsPosition is set to LEFT or RIGHT, then maxThumbRows is used as a maximum value and the actual number of thumbnail rows may vary according to the space available in the user's browser window.
If you think there may be room enough for another row of thumbnails, then the problem may be with space being reserved for the thumbnail navigation arrows and changing their position (via thumbNavPosition) might help.

I hope these notes go some way towards answering your queries.

Thank you for providing the screenshot.

In the screenshot, there really is not enough room to display a 4th thumbnail on the 1st row (so Showkase wraps the 4th thumbnail onto the 2nd row).
Here's a mock-up (click here for screenshot [Link removed.]) of how the page would look with a 4th thumbnail in place.

The thumbnails are responsive but they will reach a maximum size and will not continue to scale up to exactly fill the available page width. (They will not be scaled up beyond their actual size as this would degrade their visual quality.)
I think it would be very difficult to change the way the the gallery index page thumbnails are scaled in the Showkase source code. I expect you'd need to change several files, knowledge of PHP and CSS would likely be necessary and you may encounter problems (such as with breakpoints) along the way.
You are certainly welcome to try but I'd recommend sticking with the way that Showkase currently works.

Having said that, you could try changing the dimensions for the thumbnails (to compensate for your increased page width) via the ss_totalIndexThumbWidth value in the '_themes\base\theme.ini' file (on line 27). (Please note that the line number refers to the current version of Showkase, v1.7.2.)
This might help to space out your thumbnails a little better to give a more balanced looking page layout.
Be sure to republish your site after making any changes.

What you are describing is the Splash Page.

The Splash Page is a placeholder for the gallery which is displayed by default on small screen devices when the gallery is embedded in a web page alongside other content (rather than displayed on a page of its own with dimensions of 100% x 100%, filling the browser window) and may may too small to be usable.
The Splash Page is essentially an image link for the gallery which displays an image (by default, the first image in the gallery) and some text.
When the user clicks or taps the Splash Page, the gallery is expanded to fill the browser window (giving the images more space to be displayed).

More information about the Splash Page can be found in the Screen Modes section of the Gallery Tour.

You can choose to not use the Splash Page by setting showSplashPage="NEVER" ('Splash Screen (Pro)' section). Using this setting, the Small Screen Mode thumbnail page would initially be displayed instead of the Splash Page on small screen devices.
Alternatively, you could force the gallery to be displayed in Large Screen Mode (which, by default, does not use the Splash Page) on all devices and in all browsers by setting screenMode="LARGE" ('General (Pro)' section).

For reference, all the Splash Page configuration options can be found here.