You can use "create application shortcut" feature of chrome to create an application shortcut to gmail and use that from farr to launch gmail inbox (this is basically a shortcut to chrome with an argument of gmail.com so you can just use that in an alias or sth).
As for the feature of displaying already active gmail, or start a new window if not running you can use ahk scripts.
I am working on an idea of a farr plugin that will do just that for any app. But it will not be ready any time soon.
