Everything seems to be working: Summary
Apparently after I upgraded to ISPconfig 2.2.24, the master autoresponder.rc.master that I edited to add SENDMAIL=/usr/sbin/sendmail was overwritten but the autoresponder still worked even without the workaround. I don't know if it fixed the original issue but autoresponder is working without any modification.
I added a DNS A-record at my DNS maintainer for my server, server01.domain.com, and changed the hosts file and entries in main.cf for postfix and httpd.conf for the ispconfig web. I don't know if that's all I need to change (any ideas?). I also checked via ISPconfig control panel and made sure the DNS entries point to domain.com only for MX records.
Everything seems to be working and I do get email responses even if I send it from a mail address outside of any hosts I have on server01.
Thanks a lot for everything. Hope this helps someone else as well.