While New Outlook for Mac attempts to preserve as much compatibility as possible with GMail, there are some functions that do not work in Outlook and must be performed in GMail.
Email rules must be created in Gmail
Rules can be created and managed here: https://mail.google.com/mail/u/0/#settings/filters
or
In Gmail > Press the Settings icon > See All Settings > Filters and Blocked Addresses
Auto responder must be created in Gmail
In New Outlook for Mac, the option to create an auto responder is greyed out. The auto responder can be created and managed here: https://mail.google.com/mail/u/0/#settings/general at the very bottom of the page.
or
In Gmail > Press the Settings icon > See All Settings > General
Once the auto responder is created, press Save Changes to apply.
Known issues syncing Google accounts to the Microsoft Cloud