Yes, it will work with Google Assistant. Set it up as a smart home device in the Google Home app.
There are online best-match tools to find if it will work with your wall heater - my guess is that it won't, but one of the higher end Nest thermostats might.
The display is a muted display of two or three icons. They automatically blank after 10 seconds when you walk away.
Yes, all the features are available from the Nest app.