I work at a small phone repair chain. When I'm logged into the Gmail or Inbox apps with the company's shared Google e-mail account on my private phone, my phone is listed in Android Device Manager for the work account. This allows any and all of my colleagues to Lock or Erase my phone. No thank you.
My phone is an LG G4 running Marshmallow.
How can I remove (not hide) my phone from the Android Device manager for my work account and still receive e-mail on it?
As pointed out by OP signing out of work account signs out of the Gmail app and any other Google apps. So there is no way to receive work mails using Gmail or Inbox app
But you want to sign out to be outside the control of Android Device Manager, yet access company mails. For this, you would need to sign up on your device using your personal gmail account and use a browser for work account. You would need a browser that can send you notifications when mail is received on your work account.
Firefox can do this . Install X-notifier lite add-on to receive notifications. There are a couple of other Gmail add-ons but they did not get installed (you may like to try). This is a basic notifier that shows messages received in your notification tray
Snapshot on left shows add on showing up in Firefox menu and on right the notification appearing in your notification tray
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