I'm not sure how you have user accounts and related settings (e.g. environment variables) configured on your system, but the default install location for Steam (or really any application) on a normal Windows 7/8/10 setup is definitely not on the desktop.
That is what I was thinking too. My default was C which is my SSD. I use that for Windows so I changed it to D which is my HDD. Now all games download to D. I would think installing on your desktop would need to be picked and not something Steam (or any program) defaults to.
When it made the mess, first thing I tried was uninstall. Didn't work. So I tried to put them all into a folder and just forget about it. Didn't work. Uninstall. Remove program manager. All that. It's quite a finicky program to try and get rid of.
So you have all the folders and files in a Steam folder now? Have you tried deleting that entire folder and reinstalling?
If you right click and open properties in one of those files that installed on your desktop, what does the path say? I wonder if you installed it in your desktop folder but there is also a Steam folder on your C drive in Program Files/Steam.