Is it possible to create a custom Google Lab project? Or do Google Engineers have to create them? I'd like to integrate Gmail with another web application on a company web server.
If that is not possible, is there a way to import/export google mail in some format (ie: XML, HTML, etc..) for a specific gmail user or group of gmail users?
No.
You're looking for the IMAP protocol, which allows you to interact with Gmail messages.