MAIL: Trying to settle long-lasting problem. Mail always immediately downloads the messages I delete from any IMAP mailbox. I'm running out of HD space!

Question

BACKGROUND
I only have 60GB of space on my MacBook Air. Currently, three long-standing IMAP accounts are taking up over 7GB of it!
I have seen several related discussions about this kind of problem, but they all seem to have been solved or glossed over with setting up the IMAP interaction so that messages are deleted from the server in addition to the device or with a "Mailbox > Use This Mailbox For" fix that doesn't work for me. I like being able to save all of my email history on servers (I'm often surprised by how many years back I have to search to find a piece of information I need), but I can no longer stand losing so much of my very limited local space to really old mail.
PROBLEM
I'll refer to just one of the relevant accounts as "Gmail," since that's in parentheses when seen in these screenshots. These are its current Mailbox Behaviors settings:
Under my Gmail Inbox in Mail, I have deleted every message from before 2013. The mail from 2013 is all that I want stored on my MacBook Air.
Ultimately, I think this has something to do with the Archive (seen in the sidebar above, as well). Even when I've deleted everything from before 2013 in my Inbox, AND emptied the entire contents of Trash manually, the Archive still contains all of the mail from the server (yes, this account was started in 2011, but I have others that go back much further).
The cherry on top is that when I delete the same messages from the Archive as I've deleted from the Inbox, Mail immediately begins re-downloading all of those messages. Last time I tried this my laptop had to download 23000 messages, but here's a screenshot of proof with a smaller sample:
In case my Gmail server settings are useful to consider, here they are:
Final Notes
• There's a lot that I don't know about exactly how IMAP communication occurs between a device and a server. For my sake and the sake of anyone else who needs this kind of help, please clarify terms like "flags" and "archive" (since Gmail and Mail both use the term and I don't know think they mean exactly the same thing) if/when you use them.
• If anyone does think that there's some sort of "Mailbox > Use This Mailbox For" solution to this, you should know that no matter what I have selected in Mail's sidebar (at no matter what level of indentation), the options available under that menu item are always grayed-out.
• I want to keep using Mail. There's a lot that I like about it and I don't want to switch to Thunderbird or something else (remember, I need to save as much local space as I can).
• I just upgraded to 10.9.0 Mavericks, but this problem has existed for me and behaved essentially just like this as long as I can remember. In fact, I think I recall the problem still existing before there was an Archive in the sidebar at all.

Answer 1

UPDATE
I still have the problem, but when I checked on this post to see if anyone had commented and I simply hadn't received an e-mail, I noticed that all but the last of the images I included have stopped displaying (I had friends check, too).  Most of them were just visual examples of what I was explaining in hopes of giving context and all available info that I may not have personally viewed as relevant, but the first one, about my account's Mailbox Behaviors settings sure is important. These are the settings I have:
• Store draft messages on the server — Check
• Store sent messages on the server — Check
• Delete sent messages — One month old
• Store junk messages on the server — Check
• Delete junk messages — One week old
• Move deleted messages to the Trash mailbox — Grayed-out and Check
• Store deleted messages on the server — Grayed-out and Check
• Permanently erase deleted messages when — Quitting Mail
I hope this helps. Please comment if I'm leaving out any other important information.