Why no "Allow Offline Access" option in Default API and Invalid Token for Custom Api?

This is a setting that would grant refresh tokens for the management API for your tenant. I don’t think this is what you want to do.

I think you are looking to get refresh tokens for Google.

In order to get Google refresh tokens, include the following params when you log in with your user. For example:

GET https://YOUR-TENANT.auth0.com/authorize
  ?client_id=xxx
  &response_type=token
  &redirect_uri=http://YOUR_APP/callback
  &scope=openid%20name%20email
  &access_type=offline
  &connection_scope=YOUR-CONNECTION-SCOPES

The refresh token will not appear until you have logged in again with the user.