Renaming, deleting and moving files must be done through your cloud storage provider. You can't do this through the Document picker. So you will need to open your cloud file store - whether that's Dropbox, Google drive, One drive etc and do your moving, deleting and renaming from there.
Another way you can rename a file is when you make a copy - you can simply click on Documents > Make a copy > Same Tab/New tab and save the file in a new location with a new filename.
If you rename a file, then it will still have the old name in Recent Files until you open the newly named file. If you delete a file it will still appear in the Recent Files panel of the Document picker but you will not be able to open it - you'll receive a warning that the file couldn't be found in the cloud drive.
If you tap Re-try or Cancel the picker will refresh and the deleted doc will no longer be visible.