In Mail.app - Preferences - General, select 'Mail Notifir.app' from 'Default email reader'.
Yes! The free version is called Gmail Notifr (v0.9.0), which can be downloaded here.
The free version will not have any updates in the future.
You can either uncheck the 'Use Notification' checkbox under account detail settings, or disable or customize the notification from OS X's System Preferences - Notifications.
Yeah! But you need an App password to access your Gmail feed, follow this guide to generate an App password then use that password in Mail Notifr.
Yes. Your password is store in OS X's Keychain securely.
When Mail Notifr checks your new messages, it talks to Gmail feed using HTTPS.
There's a hidden feature to do this. In Terminal.app, use this command:
defaults write com.ashchan.GmailNotifr AutoCheckAfterInboxInterval -float 30.0
Change number 30.0 to seconds you want to delay before an extra check. The extra check will update your count number.
Set the value to 0 to disable this feature.
* Gmail™ is a trademark of Google Inc.