Re criteria for filtering stories: karma isn't a factor; past submissions are. A downside of the latter might be a rich-get-richer effect, but we mitigate that by looking for any earlier submission of the same story and picking that one instead. We don't advise about timing; in fact we randomize the process to reduce timing effects.
We mostly don't send repost invites anymore, but rather re-up the original post by rolling back its internal clock and giving it a random placement near the bottom of the front page. This guarantees a few minutes of community exposure, and regular upvoting decides the rest. On the front page, we display the re-upped timestamp, not the original timestamp, because otherwise it's confusing—only the re-upped time makes sense relative to the other stories on the page. But the original timestamp is available on most of the other pages that list the story, like /submitted and /from, and the discrepancy is temporary, since eventually the two timestamps converge.
The reason we switched from reposts to re-ups is that HN users are averse to duplication. I trust that the programmers among them hold their code to the same standard. We do still send emails when it seems important for the submitter to know that their story may still make the front page—mostly for Show HNs, like this one.
All of this is the latest in a series of experiments we've been running, whose goal is to fix the problem of good stories falling through the cracks on HN. Earlier posts about this, for anyone who wants more background:
https://news.ycombinator.com/item?id=10742610 (more recent, but might as well keep this list up to date)