Although the sitemap index file can list sitemaps in a different directory, the sitemap files themselves cannot reference URLs that are outside of the directory tree/subdirectory in which they are located - which I believe is what you are asking in the second part of your question. So, no, in answer to your overhaul question, I don't think you can. EDIT: At least according to the sitemap spec.
The only criteria with referencing sitemap files from the index is that the sitemap files are on the same domain. (By the way, the sitemap index file should have
<sitemapindex> as its root element, not
<urlset> as in your "Sitemap.xml" example - these are for ordinary sitemap files.)
Note: A Sitemap index file can only specify Sitemaps that are found on
the same site as the Sitemap index file. For example,
http://www.yoursite.com/sitemap_index.xml can include Sitemaps on
http://www.yoursite.com but not on
Ref: Using Sitemap index files (to group multiple sitemap files)
However, according to the sitemap protocol, the sitemap files themselves cannot reference URLs in a parent directory, or a directory outside of the subdirectory (as I think you are implying in the 2nd part of your question), as it is deemed that the creator of the sitemap in one location might not have permission to access URLs in an entirely different subdirectory.
The location of a Sitemap file determines the set of URLs that can be
included in that Sitemap. A Sitemap file located at
http://example.com/catalog/sitemap.xml can include any URLs starting
http://example.com/catalog/ but can not include URLs starting
Ref: Sitemap file location
This is my interpretation of the official sitemap protocol specification. However, implementation would seem to suggest otherwise - see @DisgruntledGoat's answer, but also take note of comments regarding robots.txt.