If I send an email (not a reply) using On Behalf or Send As, should I expect the originating NEW email to be in the Office 365 Group? I do not see email (not a reply) sent FROM group email address show in the Group box.
Nope. This functionality is only available when sending on behalf of/as Shared or User mailboxes, not Groups. In any case, the copy of the email should end up in the Sent Items folder, which is not exposed for Groups.