Icon on desktop

Just to add what LinSims has said, you may not see the 'Send to desktop' option when right clicking Photoshop. If you see 'Open file location', it could take you to:
C:\ProgramData\Microsoft\Windows\Start Menu\Programs

There you will see shortcuts and you can copy and paste the Photoshop one (and others) to your desktop.

Assuming you're on Windows, what you can do is click Start and then start typing Photoshop. When it shows up in the search results, right-click on it and then click Send to desktop (create shortcut).

You need an Internet connection about once every 30 days (I think it's 30) so that CC can perform a subscription check and make sure you're still a subscriber.