To show the Sent Mail folder under the account name, Tools/Account Settings, Server Settings, Advanced, set the IMAP Server directory to [Gmail], restart TB (see picture).

Thanks, sfhowes, but that moved ALL the folders from the Gmail folder up into the account folder, and eliminated the ">Gmail" dropdown; what I wanted was to move up only the Sent folder, leaving the other Gmail folders accessible from the Gmail dropdown (">" symbol). The good news is that when I cleared the IMAP server directory field and rebooted Tbird, it reverted to what I had before, namely under the account name there was only "Inbox" and ">Gmail" so at least I'm no worse off than I'd been! Thanks for trying, though!

You can have it one way or the other. you can not move the sent as you describe.

But have you checked that the sent folder in account settings > copies and folders is set to the gmail "sent mail" folder. Note the subtle difference to Thunderbird sent folder. This replaces the sent with the [Gmail] sent mail emails but does not actually remove the folder which is something you apparently for some reason do not want.

Thanks so much for replying, Matt! I have three e-ddresses accessed by Tbird, and you were right that the new one whose Sent folder I am TRYING to move had its "Account Settings > Copies and Folders" option NOT set to the Gmail folder, unlike that of the other two e-ddresses. I corrected that, closed and re-opened Tbird, hoping the Sent folder might be visible just under the account name (as the other two have long been), but it wasn't. It's that symmetry (having all three Sent folders visible under the account name) that I'm seeking. But I guess I'll just have to believe your first sentence! Thanks again for replying! Regards, Dave

Is the sent folder still showing the awful arrow looking icon beside it? if so it has not changed... what you changed was not right.

If not go back again and made sure it is set to other and the sent mail folder on the correct account. While you are there ensure archives is set to all mail on that account.

Then go to the junk setting and change the folder to spam on the gmail account instead of the default Junk folder.

Next, thunderbird does not by default synchronize the junk and trash folders unless you open them. This is all good, except you never actually see the emails Google think are spam. The false positive emails all the time in my account. Right click the folder, select properties and then select the option to check this folder when getting messages if you want to keep the folder updated. I do because then I get an updated number showing unread spam on the folder pane.

Finally. You could delete the file foldertree.json file in your profile folder while Thunderbird is not running. That will force a folder pane rebuild that will probably fix your folders if the settings are correct.

https://support.mozilla.org/en-US/kb/profiles-where-thunderbird-stores-user-data

Once again, thanks for your generosity in helping with my problem, even though I can still see the new account's Sent folder in ONLY the Gmail dropdown (not just under the Inbox under the account name, as it is for two other accounts). Perhaps those two accounts were established before some Tbird update and that folder-moving capability has been removed. I did the other changes you suggested, and am posting this only to 1) again thank you for your kindness, and 2) accompany an image of the arrow, to see if it is the "awful arrow looking icon" you mentioned. In the attached file the arrow is the same as the one shown in my new account in which the Sent folder is still under the Gmail dropdown. I HOPE that's all clear......  :-D

Well, the uploading failed because I'd pasted a screen capture of the arrow icons into a Word doc, which file I'm disallowed from uploading. The arrow is like a Christmas tree tilted maybe 30º to the right, if that describes what you were referring to.

I've wasted way too much time on your part (not to mention my own time) so you need not reply! Best regards, Dave