I was doing a normal update on the creative cloud for Photoshop and Lightroom. After the update, Photoshop disappeared from the desktop on my PC. Weirdly, it also does not appear on the Creative cloud anymore so I can not even reinstall it. I still have Photoshop CC version (round edges on the icon) but the version with sharp icon corners I can not find anywhere. It just disappeared from the desktop, bottom taskbar, Creative cloud. Quite pissed because I lost the Photoshop projects.
You haven't lost any Photshop projects. They are still there on disk, and they will still open in the new Photoshop versions.
Apparently you haven't updated in a while. The icon had sharp corners a few Photoshop versions back; in the later versions the corners are round. So you have simply been using an old Photoshop version.
As for keeping old versions, if that's what you want, you can do that by unchecking "remove old versions". As a matter of fact, keeping old versions should be the default now.