These are rather detailed instructions for what worked for me. I now can finally send securely, authenticated and reliably, in 10.10.3
Part 1: Clear out Apple Mail settings
1. Open Mail -> Preferences
2. Under "Accounts", go to the email account for which the SMTP connection can not be established.
3. Under "Advanced", un-check "Automatically detect and maintain account settings"
4. Under "Account Information", go to "Outgoing Mail Server (SMTP), and choose "Edit SMTP Server List..."
5. Under the SMTP Server List, remove the SMTP servers that are not working, by clicking the minus sign.
6. Close the SMTP Server List by clicking "OK."
7. Verify the Outgoing Mail Server (SMTP) entry is set to "None".
8. Close Apple Mail
Part 2: Clear out Keychain settings
1. Open the app "Keychain Access"
2. Choose the "iCloud" Keychain in the left menubar
3. Click on "Kind" the list of entries by kind
4. Carefully go through the list of items of kind "Internet password." Go through the list one by one, and look for the ones that have "Where: smtp://..."
5. Remove internet password entries that are SMTP connections to the server you cannot connect to.
6. Do this slowly so you don't accidentally remove wrong entries.
7. Close Keychain Access
Part 3: Start with a clean slate
1. Restart the computer
2. Close down virus checking software
3. Close down any running apps for monitoring connections, such as Little Snitch and Hands Off
Part 4: Re-establish an SMTP connection
1. Open Apple Mail
2. Go to Apple Mail -> Preferences -> Accounts again
3. Click on the account for which SMTP is not functioning
4. Under "Account Information", go to "Outgoing Mail Server (SMTP), and choose "Edit SMTP Server List..."
5. Create a new SMTP server by clicking the plus sign.
6. Under "Advanced" un-check "Automatically detect and maintain account settings.
7. Type in the SMTP information that you know should work.
8. Press "OK"
9. Verify "Outgoing Mail Server (SMTP)" has selected the correct server.
Part 5: Keychain entry
1. A pop-up window pops up, asking you to trust a certificate.
2. Carefully note down the name of the certificate. In my case, the certificate was issued by Comodo
3. You must trust this certificate.
4. Open Keychain Access
5. Find the new certificate under the "Certificates" menu.
6. Click on the certificate
7. In the certificate, click on the "Trust" triangle to open the trust section.
8. Set the following three entries to "Always Trust": "Secure Sockets Layer (SSL)", "Secure Mail (S/MIME)", and "X.509 Basic Policy". Leave the other entries at "no value specified"
9. Close the certificate window
10. Close the Keychain Access app
Part 6: Check if it works in Apple Mail
1. Open Apple Mail
2. Open Window -> Connection Doctor
3. Mine finally showed "Connection and login to server succeeded." Hurray.
Key points:
* Make sure mail settings are not synchronized
* Don't run Little Snitch or Hands Off while you set this up
* Use both Apple Mail and Keychain Access app while setting this up