Asterisk does support Automated Speech Recognition in a very cost effective way with the module from Vestec.
Cepstral Text to Speech functionality works great with Asterisk as sounds almost lifelike (pick Alison would be my advice)
You have me on MRCP though, I dont know what that is.
Given that Asterisk can store its voicemail in an imap store, it'q quite feasible to use Zimbra to hold that, but you would likely need to use a forumalted password that each system knows. That way the Asterisk server would authenticate the caller by DTMF and then signin to the Zimbra server using a stored or calculated password, there's no need to the caller to be able to, or authorised to, access the Zimbra server directly.
I dont know if any of that helps though, or even if it begins to answer your initial question! If I'm way off Im sure someone else will chime in and correct me soon
