I'm also having a suspicion that the [-] button on the right, which I've been using to collapse threads, might be the one. Either way, I find HN's UX confusing.
You need x karma to see it.
The [-] collapses the subthread, nothing more.
Even past that threshold, you can't downvote responses to your own comments. That's probably a wise limitation when you think about it.
Downvoting threads is done with the "flag" button which appears at the same time. Unfortunately, as HN has grown past a certain size, there's gotten to be a critical mass of "made" high-karma censors, who flag nascent postings on a whim. The diversity of articles suffers, limited now to either the things censors like, plus a small selection of postings where teams will "unflag" (also available at 501) this price-of-admission flagging.
Running any kind of good forum is hard, and HN's moderators work really hard at it, but "strangled-in-the-crib" flagging both takes away from the value of the site and gamifies the forum to where it now takes a team to make a posting that survives. I don't have an easy solution, though it might be worth trying to limit flags to one, two, or ten per day.
I think your idea of limiting the number of flags a user has per day is a good one. I'm kind of surprised that isn't the default behavior already...
I have over 1,800 points, and there's no downvote to be seen.
I do see downvote on some comments (not all).
The rules seem subtle.
Submissions don't have downvoting, but you can flag them.
Don't like a post? Just ignore and don't upvote it!
Sadly, votes are too easily mapped to agree/disagree or truth/falsehood - see the "1,800 karma" comment below as a key example, where it furthers the conversation but is downvoted "because they are incorrect".