Heh, it's funny, while I was gathering information to reply you, markc has been replying too
First of all, Gmail accounts do work wonderful at Feng Office.. but yours happens to be a Google Mail one, so I assume you might be from the UK and I'm not so sure why do you have this issue
So, you have already mentioned it, but could you please double check if your IMAP settings are enabled for your account?
The other thing I was going to suggest you to try was using imap.gmail.com instead of imap.googlemail.com as II'm not so sure what's Google Mail default email server, but it seems like you have already tried that, otherwise please do so.
What about the POP3 settings? Have you tried that way if you manage to connect to the server?
Regarding the calendar synchronization: you may synchronize your calendar from Google Calendar to Feng`s, but not viceversa
About the contacts, you may export all your contacts either from your email account and then import them to Feng Office, and viceversa
Greetings!