To move a file to a different folder in the Files app, simply deep press (or long press) it until a menu pops up. Select "Move" to bring up a directory tree. Tap on the folder you want to put it in then tap Copy in the top right corner.
To change the name of a file in your Files app simply deep press on it (or long press) until an options menu pops up. Choose Rename and type in the new name.
Notes was the first app to feature a scan document option, which debuted with iOS 11. iOS 13 has added that feature to both Mail and Files. It makes the most sense for Files to have a scanner, since you can attach documents from Files to both Notes and Mail, and it gives you a place to store the scanned document if it is something you'll want to access again in the future.
Here's how to scan a document from the Files app on your iPhone:
iOS 13 finally adds support for zipping files and unzipping compressed files. You could do this before with third party apps but it's about time native support was added, especially since they introduced a file management system (Files) in iOS 11. Here's how to open and extract zipped files and how to compress your own files and folders on iPhone or iPad running iOS 13 or better:
iPhones and iPads these days have quite a bit of storage, so you may have plenty of room to save documents on your device rather than spending money to upgrade your iCloud storage to the next tier. To do this you will want to set your internet downloads to automatically save locally, which is easy to do with Safari's new download manager and you will want to make sure you save email attachments locally. Also, anything you have stored in iCloud you can always move to a local folder in the Files app.