It's the standard default IMAP arrangement. Thanks for your interest and attention.
By the way, I have some work-arounds. One work-around is to go into the GMail browser interface, where I can clearly see all the messages sitting there in the inbox, and there I create a brand new label (can't be one I've ever used before) such as "primary" or "skeezix" and then label the messages in question with it. This will force Microsoft Maill App to "re-see" the messages in question, at least in so far as I can find the new folder (same name as the new label) and then look in that folder for the missing messages. It doesn't force Mail App to "re-see" the messages fully; they still won't appear in the In-Box nor will they appear via a search through the "all mail" label for unread messages. A second work-around is to have the good sense, whenever in the Mail App, to NEVER EVER change the "folder" that a message is stored in. Don't hit "Archive" (this moves things to "deleted/archive" or to "archive" or to "archives" depending on the phase of the moon?), don't drag-and-drop to a new folder (drag-drop will remove FROM the old folder, but not place INTO the new folder, thus accomplishing the disappearment of the message(s)). Third work-around is to find unread messages, but you have to use a trick to find them. Mail App won't find them properly. Instead, you must search in "all mail" for "is:unread" text string. This trick alone will find unread messages. I can't just use the Mail App's "unread" filter, that won't work. I have to use the text string "is:unread" to find unread messages. The contradiction in "unread" status -- Mail App won't even find NEW messages that are unread, if I just use its own "unread" filter -- indicates (I think) that GMail puts an "unread" LABEL on the message, rather than actually changing its STATUS as un/read, whereas MS Mail App cannot understand LABELS. So, again, the culprit seems to be, that "folder" (all other sane decent emailers) versus "label" (GMail alone) is inconsistent.
Your thoughts are welcome. In fact, at this moment I don't actually HAVE Outlook (the software) -- I have an outlook-dot-com email account, but I don't have rights to the Outlook desktop app. This is why I was hoping to use MS Mail App, to replace Outlook desktop software. I've tried Thunderbird, there's really no other alternatives. I am in process of transition from a full-scale business-level Office 365 account (at over $150 per year ouch) which was fully hosted at my-personal-name-dot-com, to nothing but email forwarding. I had thought I would transition over to GMail for email purposes, but this experience has pretty much made clear that's not a good solution. I wanted to avoid web-based email work (I hate both the Outlook and the GMail browser experiences) but right now I can only "fix" GMail's MS Mail App problems by resorting to GMail's web interface, so that means MS Mail's purpose is defeated.
I've managed to set up MS Mail App with TWO gmail accounts, identical email address. One of them only has email (set up via "other" method and IMAP details), the other has all three of email-calendar-contacts (set up via automatic "Google" method). The two accounts, identical email address, identical software running at the same time, show different in-box content. If you'd like I can tell you a zillion ways to make a GMail message disappear and remain invisible reliably ...
