Most of the problems in this issue were solved already so it could almost be closed:
* patch 1 was addressed in #11927
* patch 2 was addressed in #4066
* patches 3 and 4 were addressed in #11893

Torsten's problem was addressed by bcf04ced5ef1.

> I'm not sure what is left to do in this issue.

The only patch remaining is patch 5. I attached an updated version against tip of default branch. My patch mimics shutdown in imaplib.py in that it silences ENOTCONN. However I don't have a test that fails without the shutdown and I don't know if checking ENOTCONN is really needed. I tried to get shutdown to produce ENOTCONN by using Postfix as a server with smtpd_timeout=5s, connecting to it and waiting idle for more than 5 seconds before doing close(). In the Postfix logs I see that Postfix disconnects after 5 seconds of inactivity but doing shutdown afterwards doesn't trigger any exception so the ENOTCONN part remains unexercised.

My patch also adds shutdown method and SHUT_RDWR constant to mock_socket.py since otherwise test_smtplib fails.

(Added Antoine to nosy because he reviewed the patches for #11927 and #11893)