Yes, but only for comments. And it's limited not only by karma, but time: It's only available for a rather short time after a comment was posted. Dunno how long exactly, but feels like something between... 12 and 24 hours?

For comments, yes, for stories, only flag but not downvote.

But both flag and downvote are karma-gated (I think the threshold is 500 for both, but I could be wrong.)