I first tried POP3, but for some reason, Thunderbird would not delete
the mail from the Gmail server when I deleted it locally (even though
I had that option checked).  Also, I used the Advanced Config in
Thunderbird to try and get it to delete the messages when I move them
(mail.pop3.deletefromserveronmove = true).  That wouldn't work
either.

So, now I'm onto trying IMAP.  And, I'm running into similar, but
different problems.

I'm trying to move messages from my Gmail Inbox into a Local Folder
(Local Folders/Whatever) in Thunderbird.  My intent is to remove the
email from Gmail server entirely and have just a local copy on my PC.
What is happening is I get a copy in Local Folders, but the same email
also still exists in the Gmail All Mail folder.

I saw someone mention in these threads that to accomplish this, you
need to move the mail from the [Gmail]/All Mail folder to the Local
Folder, but when I do that, I get a copy in the Local Folder, and the
mail re-appears in [Gmail]/All Mail.  I can repeat moving from [Gmail]/
All Mail to Local Folder, but then I get two copies in Local Folders,
and then it re-appears in [Gmail]/All Mail.

Is there any way from within Thunderbird to move a mail from Inbox (or
[Gmail]/All Mail) into a Local Folder and have that email deleted from
the Gmail server?

Secondly with IMAP, I get a similar circular problem when attempting
to delete mail from within Thunderbird.  When I delete in Thunderbird,
it puts it in Trash (not [Gmail]/Trash).  So, Gmail actually labels it
as [IMAP]/Trash.  When I tell Thunderbird to empty the trash, Gmail
removes the label, and the mail still exists in All Mail.  So, the
only way to delete a mail is to go to [Gmail]/All Mail, and drag the
mail into [Gmail]/Trash.  This is a pain when I'm trying to read
messages from Inbox.

Or, any ideas on my POP3 delete mail problem?  (I could go back to
POP3 since IMAP is not working as expected)