Sorry, I've been offline for a few days and am only around for a short while now.
Where exactly are you seeing this "data recovery all files" folder? Because I've looked at all of your screenshots and see no such thing in any of them. Hence I'm quite confused as to where this is coming from. Is it the name of your unremovable shortcut on the home screen? If so the problem is that could be anywhere: the filesystem listing you've shown (output of "ls -al") is just showing the top level directories of the filesystem, but the file or directory you are after could be deep in one of those for all I know). So can you tell us where it is, or post a shot that shows it?
As a matter of curiosity, is your device rooted? I'm surprised you could produce that filesystem listing if it isn't.
I'm pretty sure that you could remove the shortcut from your desktop by clearing the data for your launcher app. You'd have to set up your homescreens again, but the unwanted shortcut would be gone. As for the file or folder itself, I'd need to know where it actually is. Changing its file permissions (the "chmod" command) requires you to either issue the command from within the same directory as the file/directory or to specify the full path. I'll try to illustrate this with an example: suppose I want to modify the permissions of a file called "junk" in a subdirectory called "FileRecovery" inside the "sdcard" directory, then I could either do:
chmod ugo+rw /sdcard/FileRecovery/junk
(which gives user, group and other read and write permissions to the specified file or folder)
or
cd /sdcard/FileRecovery
chmod ugo+rw junk
(change directory to where the file or folder is, then change its permissions)
Note that if this is an empty directory you can delete it by typing:
rmdir junk
(or rmdir /sdcard/FileRecovery/junk).
As long we you have write access to that file/folder. For an ordinary file just use "rm" (short for "remove"). You can also zap a directory which isn't empty using the command
rm -r junk
which will delete the folder and all of its contents. Use this command with care, as you can easily delete a lot more than you intended (and if you are used to Windows and trash/recycle bins, forget all of that right now: if you delete something using the command line it is gone, there and then, no second chances. In fact if you delete anything in Android this is true no matter how you do it).
One thing to consider: I do not know whether deleting the file or folder will remove the shortcut from your desktop. I've never placed a file shortcut on an android desktop, and I've never used the particular launcher app that is providing your desktops, so can't be certain how it will behave.