I have been happily using LR6 and Photoshop Elements 13 for years. I just decided to subscribe to the current LR/Photoshop package, and installed yesterday. New LrC 10 and Photoshop are working fine. However, my old versions of LR and PSE no longer open. Clicking the icon to open them does absolutely nothing. No errors or anything. Just no response. I would assume the old versions of my software are still installed and should still operate. Windows 10.
With the Plan that includes Lightroom-Classic and Photoshop, I should think there would not be any need to use LR6 or PSE.
Icons may be a shortcut to Programs or Catalogs or files, and a [Right-click] to show their 'Properties' may indicate their 'Targeted' function.
Lr-Classic will upgrade your LR6 Catalog to look identical, and give you all the new features added since LR6. And you canot use the new V10 Catalog in LR6- Not backwards compatible. No point in trying to 'go back'! ANd an icon that points to a v6 Catalog should be deleted!
The install of Photoshop may have changed, the icons to start PSE, but it should not un-install PSE, even if the icons were deleted. Look in Windows: Press the [Windows] key and type "Apps"- you should see PSE still listed.
Many people have Photoshop as the 'default' external editor in Lr-Classic, and also have PSE-Editor as an "Additional External Editor" defined in the Preferences.