Hi,
This sounds like normal behavior,
It should allow you to send to the domains you have configured it as a backup MX for, but not for any other domains, otherwise it would be an open relay.
If you want to send mail using this server you should configure the mynetworks parameter.
as for zimbra starting on boot, this is configured with run levels, (
Runlevel - Wikipedia, the free encyclopedia)
look at what distribution you are using, and then remove the S99zimbra symlink from the appropriate run levels
Make sure you aren't using the zimbra version of postfix/amavisd though, those will stop working too if you remove the S99zimbra link.
Regards
Jeroen