****%&** !!!
Nothing. I've tried all the steps above. I've tried 2 configurations of Username (I ought to have been clearer in my question. My AppleID here on discussions is 'willingfoe'. My AppleID for iCloud is 'user.name@me.com'. I think it may be possible for someone to set up an account where the username doesn't include the '@me.com' portion - for iCloud, my username does include the @me.com). Does the 'username' you use above end in '@me.com'?
Interestingly, Mail checks and receives incoming mail whether I have only put 'user.name' in or the full 'user.name@me.com'. The 'Trash' folder is functioning and deleted items disappear from the 'Inbox' and show up in the folder in all devices regardless of where they're deleted from.
If I open the 'Account Info' window and select the account, the Quota Limits are displayed, including for the 'Sent' folder (hovering the mouse over this in the Mail sidebar says 'Contents stored on server. Name on server: Sent Messages'). Thus, lots of the queries and so on currently being handled without issue but I have a message stalled in my outbox.
I ran Connection Doctor a few times (lots of deleting from SMTP lists and keychains in the meantime). The first time, from what I could see, it was trying Port 25. I deleted the old one and set it specifically to 587. Running Connection Doctor, here are the 2 relevant sections - this is way past my level of experience but I think it might be helping in at least diagnosing the problem.
For the Incoming Server (I have substituted ••••••.•••••• for my Username, ********** is how the password is displayed to me in Connection Doctor):
CONNECTED Oct 23 15:07:37.836 [kCFStreamSocketSecurityLevelNegotiatedSSL] -- host:imap.mail.me.com -- port:993 -- socket:0x116269da0 -- thread:0x1162b8970
READ Oct 23 15:07:38.199 [kCFStreamSocketSecurityLevelNegotiatedSSL] -- host:imap.mail.me.com -- port:993 -- socket:0x116269da0 -- thread:0x1162b8970
* OK iSCREAM ready to rumble (r1F3601+:14874)
WROTE Oct 23 15:07:38.211 [kCFStreamSocketSecurityLevelNegotiatedSSL] -- host:imap.mail.me.com -- port:993 -- socket:0x116269da0 -- thread:0x11625bbb0
1.29 CAPABILITY
READ Oct 23 15:07:38.331 [kCFStreamSocketSecurityLevelNegotiatedSSL] -- host:imap.mail.me.com -- port:993 -- socket:0x116269da0 -- thread:0x11625bbb0
* CAPABILITY st11p00mm-iscream004.me.com r1F3601+ XAPPLEPUSHSERVICE IMAP4 IMAP4rev1 SASL-IR AUTH=ATOKEN AUTH=PLAIN
READ Oct 23 15:07:38.347 [kCFStreamSocketSecurityLevelNegotiatedSSL] -- host:imap.mail.me.com -- port:993 -- socket:0x116269da0 -- thread:0x11625bbb0
1.29 OK !!
WROTE Oct 23 15:07:38.364 [kCFStreamSocketSecurityLevelNegotiatedSSL] -- host:imap.mail.me.com -- port:993 -- socket:0x116269da0 -- thread:0x117313a60
2.29 LOGIN ••••••.•••••• **********
READ Oct 23 15:07:38.556 [kCFStreamSocketSecurityLevelNegotiatedSSL] -- host:imap.mail.me.com -- port:993 -- socket:0x116269da0 -- thread:0x117313a60
2.29 OK User ••••••.•••••• logged in 713339
WROTE Oct 23 15:07:38.575 [kCFStreamSocketSecurityLevelNegotiatedSSL] -- host:imap.mail.me.com -- port:993 -- socket:0x116269da0 -- thread:0x101311360
3.29 CAPABILITY
READ Oct 23 15:07:38.707 [kCFStreamSocketSecurityLevelNegotiatedSSL] -- host:imap.mail.me.com -- port:993 -- socket:0x116269da0 -- thread:0x101311360
* CAPABILITY XAPPLEPUSHSERVICE IMAP4 IMAP4rev1 ACL QUOTA LITERAL+ NAMESPACE UIDPLUS CHILDREN BINARY UNSELECT SORT CATENATE URLAUTH LANGUAGE ESEARCH ESORT THREAD=ORDEREDSUBJECT THREAD=REFERENCES CONDSTORE ENABLE CONTEXT=SEARCH CONTEXT=SORT WITHIN SASL-IR SEARCHRES XSENDER X-NETSCAPE XSERVERINFO X-SUN-SORT ANNOTATE-EXPERIMENT-1 X-UNAUTHENTICATE X-SUN-IMAP X-ANNOTATEMORE XUM1 ID IDLE
READ Oct 23 15:07:38.746 [kCFStreamSocketSecurityLevelNegotiatedSSL] -- host:imap.mail.me.com -- port:993 -- socket:0x116269da0 -- thread:0x101311360
3.29 OK Completed
WROTE Oct 23 15:07:38.783 [kCFStreamSocketSecurityLevelNegotiatedSSL] -- host:imap.mail.me.com -- port:993 -- socket:0x116269da0 -- thread:0x101311360
4.29 LIST "" ""
READ Oct 23 15:07:38.914 [kCFStreamSocketSecurityLevelNegotiatedSSL] -- host:imap.mail.me.com -- port:993 -- socket:0x116269da0 -- thread:0x101311360
* LIST (\Noselect) "/" ""
4.29 OK Completed
WROTE Oct 23 15:07:38.928 [kCFStreamSocketSecurityLevelNegotiatedSSL] -- host:imap.mail.me.com -- port:993 -- socket:0x116269da0 -- thread:0x1162c18e0
5.29 LOGOUT
And for the Outgoing Server:
CONNECTED Oct 23 15:07:38.828 [kCFStreamSocketSecurityLevelNone] -- host:smtp.mail.me.com -- port:587 -- socket:0x116207210 -- thread:0x115a54330
READ Oct 23 15:07:38.946 [kCFStreamSocketSecurityLevelNone] -- host:smtp.mail.me.com -- port:587 -- socket:0x116207210 -- thread:0x115a54330
220 st11p00mm-asmtp003.mac.com -- Server ESMTP (Oracle Communications Messaging Server 7u4-23.01(7.0.4.23.0) 64bit (built Aug 10 2011))
WROTE Oct 23 15:07:38.962 [kCFStreamSocketSecurityLevelNone] -- host:smtp.mail.me.com -- port:587 -- socket:0x116207210 -- thread:0x115a54330
EHLO [192.168.1.16]
READ Oct 23 15:07:39.075 [kCFStreamSocketSecurityLevelNone] -- host:smtp.mail.me.com -- port:587 -- socket:0x116207210 -- thread:0x115a54330
250-st11p00mm-asmtp003.mac.com
250-8BITMIME
250-PIPELINING
250-CHUNKING
250-DSN
250-ENHANCEDSTATUSCODES
250-EXPN
250-HELP
250-XADR
250-XSTA
250-XCIR
250-XGEN
250-XLOOP 9C7468C76FA3C4329E36A16D156E8021
250-STARTTLS
250-AUTH PLAIN LOGIN
250-AUTH=LOGIN PLAIN
250-NO-SOLICITING
250 SIZE 0
WROTE Oct 23 15:07:39.096 [kCFStreamSocketSecurityLevelNone] -- host:smtp.mail.me.com -- port:587 -- socket:0x116207210 -- thread:0x115a54330
STARTTLS
READ Oct 23 15:07:39.211 [kCFStreamSocketSecurityLevelNone] -- host:smtp.mail.me.com -- port:587 -- socket:0x116207210 -- thread:0x115a54330
220 2.5.0 Go ahead with TLS negotiation.
WROTE Oct 23 15:07:39.850 [kCFStreamSocketSecurityLevelTLSv1SSLv3] -- host:smtp.mail.me.com -- port:587 -- socket:0x116207210 -- thread:0x115a54330
EHLO [192.168.1.16]
READ Oct 23 15:07:39.966 [kCFStreamSocketSecurityLevelTLSv1SSLv3] -- host:smtp.mail.me.com -- port:587 -- socket:0x116207210 -- thread:0x115a54330
250-st11p00mm-asmtp003.mac.com
250-8BITMIME
250-PIPELINING
250-CHUNKING
250-DSN
250-ENHANCEDSTATUSCODES
250-EXPN
250-HELP
250-XADR
250-XSTA
250-XCIR
250-XGEN
250-XLOOP 9C7468C76FA3C4329E36A16D156E8021
250-AUTH PLAIN LOGIN
250-AUTH=LOGIN PLAIN
250-NO-SOLICITING
250 SIZE 0
WROTE Oct 23 15:07:39.982 [kCFStreamSocketSecurityLevelTLSv1SSLv3] -- host:smtp.mail.me.com -- port:587 -- socket:0x116207210 -- thread:0x115a54330
AUTH PLAIN (*** 44 bytes hidden ***)
READ Oct 23 15:07:40.166 [kCFStreamSocketSecurityLevelTLSv1SSLv3] -- host:smtp.mail.me.com -- port:587 -- socket:0x116207210 -- thread:0x115a54330
535 5.7.8 Bad username or password (Authentication failed).
WROTE Oct 23 15:07:40.183 [kCFStreamSocketSecurityLevelTLSv1SSLv3] -- host:smtp.mail.me.com -- port:587 -- socket:0x116207210 -- thread:0x115a54330
QUIT
The penultimate WROTE section here is where it looks like the Server is expecting login information and all that's being conveyed is 'AUTH PLAIN (*** 44 bytes hidden ***)', resulting in 'Bad Username or Password'. Could I ask you to run Connection Doctor and see how that query is handled for you? I will of course understand if this is too much effort. Thanks for all the attention/help thus far.