Hello,
I don't understand the first part of your question: Actually, the "docker logs" mechanism works fine with Orthanc and Docker.
More precisely, if I start Orthanc with the following command (note the "--name" option):
# sudo docker run --name orthanc -p 4242:4242 -p 8042:8042 --rm jodogne/orthanc
Orthanc will write its log to stderr, which I can access using your command:
# sudo docker logs -f orthanc
Regarding the second part of your question, check out the "--logfile" and "--logdir" command-line options of Orthanc (documentation is available as usual by running "Orthanc --help").
If using the "--logfile" command-line option with Orthanc >= 1.1.0, you can send a SIGHUP signal to Orthanc, which will flush the log file onto the disk. As with most UNIX services, you can use this SIGHUP mechanism in conjunction with the logrotate standard tool running in a daily cron job to avoid the log growing indefinitely.
HTH,
Sébastien-