Hi DiZoe, thanks for your help again. I will try to answer your questions. (I just posted a long reply and this site wouldn't load it and I lost it :( )
<i>Did this issue happen after any event in particular? (E.g. update, an installation, account change)? But worked normally before? </i>
Not that I'm aware of. I first became aware of a problem of mail from my Wordpress site going to Spam. I have now solved that (I think) by getting my ISP to send Wordpress emails out under the ISP's SMTP. It was then that I noticed this issue of some incoming mail only appearing in All Mail. Just as was typing my previous reply, an email came in from my Wordpress and it didn't go to spam, but that was maybe because I have a rule to stop that. But it didn't appear anywhere else, just in All Mail. I think the two problems occurred at the same time but I can't be sure. I certainly only noticed them around the same time.
<i>And just to confirm, in Apple Mail, your other email accounts work normally, is that right?</i>
Yes, mostly. I currently have an issue with being unable to mail from my ISP's email accounts because of some server issue (the error message says "The server response was: SMTP-AUTH is not allowed on port 25" ) but I can send out on our other (shared) Mac without a problem (that Mac isn't connected to this Gmail account).
<i>What do you see in Mail > Preferences > Accounts > Mailbox Behaviors, for Gmail?</i>
Enable this account. Tick
Status: Online
Description: my gmail address
Email address: My nae and Gmail address
Download attachments: Recent
Send large attachments with Mail Drop: Not ticked.
<i>Also in Mail Preferences, are there any local “Rules” set up there to change the mailbox behavior on your Mac?</i>
Rules for the three websites I manage, but none for Gmail.
<i>And finally, (If you have the time to do so), If you create a new Admin account on your Mac, sign into that, add your Gmail account, does the same behavior occur?</i>
I haven't done this for several reasons, but I will if it is necessary.
<i>If you have an iPhone, does it act the same way as well?</i>
I don't have an iPhone.
<i>From what you said, the Gmail web site acts as you would normally expect it to is that right?</i>
I think so. But since I usually access it via Mac Mail, I don't really know.
<i>Those answers may help to isolate a cause and effect and hopefully get a solution!</i>
I hope so. Thanks so much. I still think this may be a Gmail issue, so maybe I should go back to the Gmail forum? But I think I am getting a more thorough look from you than I might get there, so thanks again.