Re: What are the characteristics of a structure?
You can delete the structure at: tiki-admin_structures.php
And choose (if your user have enough pems) to delete the page from the structure definition, or also delete the page itself.
So if you delete the structure, but not the pages, then you would be as you are looking for.
However, keep in mind that even if you have all pages in a structure (if you have them as such, it depends on what you did), you can have several other structures with only some (or none) of the previous pages. I.e., a page can be in several structures, and also access out of the structure.
See the Documentation for more information.
And last, and very important, could you please document how did you do the migration from your MediaWiki to Tikiwki?
Many many users are willing to do that, and your documentation will help them a lot to get started (or get the job done in less time & effort that hey expect?)
Cheers, and welcome to Tikiwiki Community! 
;){kind=link}
;){kind=link}