New answers tagged gmail
It won't integrate, no. I believe that it will be the default for mailto: links, e.g. mailto:email@example.com (it doesn't work here, but if you click on a link like that it will open in the default mail app). The web-app is simply Gmail in browser. If you want your emails integrated, you need a IMAP account on thunderbird, or some other email reader.
It seems that your postfix tries to issue AUTH before issuing STARTTLS (-> it does not issue STARTTLS), Have you restarted postfix daemon after adding smtp_use_tls=yes? http://mhawthorne.net/posts/postfix-configuring-gmail-as-relay.html
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