How do I get the Master of Orion 2 icon and name to appear on my main screen so I can go right into the program?

When using the offline installer, it will be automatically generated. When installing with Galaxy you have to tick an option to generate it, but in the game folder there is the shortcut for it,just copy it on desktop.

Find exe file in install directory, right click hold and drag to desktop, release and select create shortcut here.
Or delete numpty client, download offline installers and install from there, should create a link automatically.

Or since MOO2 is a DOS game, you could use one of many DOSbox launcher/organizers.