Note that since they also offer a hosted solution we should use that to play with instead of trying to install it at this point.
Integrating the issue tracker looks like it's just a call to this API: http://doc.gitlab.com/ce/api/issues.html#new-issue. I don't normally do web development myself so I'd rather not figuring out how to setup a copy of the website to hack on, but if no one else wants to try it I can take a stab at it.
Presumably mirroring our git repository would work the same as it does for mirroring to GitHub. My guess is that would be enough to get the basic git/issue tracker integration working.
One of the issues identified with Github is that it is closed and commercial which goes against the expressed desires of the PostgreSQL community. As such, it's important to note that Gitlab seems to be in the "freemium" model with a "community" and an "enterprise" version so for comparison purposes we should only look at the features in the open-source community version: