I know you want to be a team player - but look at it from another angle.  If you are not responsible for the work of that department and do make a error - you will be hung out to dry.  As for phone calls, transfer to voice mail.  Explain politely that this is not any area that I am familiar with and I wouldn't want to respond with misinformation.  I will be happy to give msg to so and so.  Send an email with the message (if not transferred to voice mail) and cc your boss.
As for the manager not responding, I would take it to my next level.  I was requested to do X, Mgr Z will not assist is completing this function what do you want me to do - leaves it in your mgr's court to address the project/issue.
I've been in similiar situation where the mgr did not respond to any below his level - you just have to find a work around.