I doubt there are clear definite criteria at all anymore. Noone is counting the number of questions/answers/users or the average score of posts. Rather than that, SE employees will take a glance at the general situation on the site, like
- Are there many good questions posted and do they get good answers?
- Are commited users staying and contributing beyond the initial hype on the first 1-3 days?
- Is meta discussion going, are the users actively shaping the site and its scope?
- Does it look like the site can keep up with being opened to the public?
Those are all more fuzzy criteria judged on a "you know it when you see it basis" and less based on clear numbers. Usually you either have the site fail miserably or look promising, there's not that much happening between that. Besides that, the criteria for how a site survives in public beta have been largely broadened recently, too, so even if there's still a few doubts about the site's progress, I'd think SE would rather expand private beta than kill the site outright. Those 7 days in that message you mentioned aren't even accurate. Last private betas I've seen went on for at least 2 weeks, if not 3.
As to the next step after private beta, that will be public beta. Prepare for being for a long time in that phase, probably years. When the site reaches sufficient activity (defined as 10 questions per day, stable), it will get graduated. But after the recently refined graduation process, staying in beta is much less of a stigma and there is nothing wrong with a site that stays in public beta forever (in fact, they're even considering getting rid of the "beta" label altogether in that stage).
As to this site, if it goes on as it currently does, I doubt there would be any reservations to transitioning into public beta.