iOS doesn't have a user accessible file system like the Mac. At least, the SDK and rules of engagement limit app developers from accessing the file system in traditional laptop\desktop manner. Thus, you cannot choose where you save docs, they go with each app.
Under the hood, each app is likely stored in /Applications and the settings for each app in ~/Library (user's library folder). Likely, the files downloaded by apps like Dropbox are stored in ~/Library/Application Support/Dropbox.
iCloud docs on the Mac are stored in ~/Library/Mobile Documents/com.apple.CloudDocs, and likely in a similar location on iOS.
iOS generally doesn't use a user profile like Macs, but the ~/ is likely for "default" user as the /Library is intended for system level stuff. Since iOS supports limited profiles for education and such, there probably is a /Users path. But, since the only way to access files stored on the iOS device is through the respective apps, such as the iCloud "Files", Dropbox, etc, you are generally limited to this method.
While it is possible for apps to share locally stored documents, this is done through the sharing mechanisms the OS makes available to apps. So, a PDF stored in Dropbox might be shared from within the Dropbox app to messages, mail, or other apps. But, other apps don't have direct access to the file stored in the Dropbox app.