Display a different greeting message to your visitor depending on which site they are coming from.
Yes it does! Just make sure that you enable cache compatibility in WP Greet Box advanced options and clear your cache afterwards.
Yes it does (even when cache compatiblity mode is enabled)!
Yes! You can easily do this on the WP Greet Box options page. You can even modify preinstalled messages or delete them.
Please take a look at more documentation available on the plugin page to see if any of them can help you.