Send/Receive Difficulties with Mail.app/.Mac Webmail

Question

I had sent a short test message to myself, using a .Mac account on Mail.app, to see if I could duplicate an error reported by another user. That message 'black holed' itself, leading to the following series of similar messages and results. For purposes of explanation:
a) messages were sent on the same account from either Mail.app or .Mac Webmail
b) "OK" means that messages showed up normally in Sent boxes and in Inboxes in both locations.
c) "Black Hole" means that messages showed up in Sent boxes in both locations, but showed up nowhere else. Ever.
d) "Junk" means that the message was sent, showed up in Sent boxes, but was slotted into Mail.app's Junk ƒ. In all cases, those messages were then moved to the Inbox and marked as "Not Junk".
Message results:
#1 — from Mail.app — Black hole
#2 — from Mail.app — Junk
#3 — from Mail.app — Black hole
#4 — from Webmail — Junk
#5 — from Webmail — Black hole
#6 — from Mail.app — Junk
#7 — from Mail.app — Junk
#8 — from Webmail — OK
#9 — from Mail.app — Junk
#10— from Webmail — OK
#11— from Mail.app — OK
Two questions or concerns:
a) 3 of 11 messages, from both Mail.app and Webmail, simply disappeared. They have message ID's, are present in the sent folders, but never showed up.
b) Mail.app's Junk filtering insisted, for a long time, on slotting those incoming message to the Junk filter (and still does, btw, on an intermittent basis). No changes have been made to Junk preferences or setups and this had not occurred in the past.
Would appreciate some of the mail gurus taking a look at this and seeing if they can find any common threads or causes. There may be more than one thing in play here, of course. TIA for any assistance.

Answer 1

I'm curious about the Junk (result "d" & question "b") aspects of your tests. Can you confirm that:
1) You do not have any custom rules that use the "Message is Junk Mail" test -- IOW, that the Junk Filter is the only rule that is using this test?
2) The messages to yourself all were sent to a "self" address that is in your Address Book app (typically, but not necessarily, the "me" card)?
3) For all/some result "d" messages they were marked as Junk?
I ask because I have intermittently seen Mail apparently ignore other default tests in the Junk Filter (particularly "Sender is not in my Address Book" when the sender is me) & evaluate the 'junkiness' of messages just on the "Message is Junk Mail" test. It has been difficult for me to evaluate David's methods for correcting this in my case because of the extremely intermittent nature of the behavior: in some cases new "me-to-me" received messages are immediately marked as Junk after removing the files David indicates, in others a few weeks later one or two are while others aren't.
In every case (so far) that some are identified as junk, training with the "Not Junk" button on a couple of messages seems to end the occurrences, as if the LSM filter has to learn they are not junk independently of the other tests.
Because of this, I would appreciate your posting to this thread any related results that occur long after you try David's fix(es), if you do try them, even if they are rare or inconsistent.
Thanks.