Do the dance pros choreograph the dances themselves each week?
Yes, with help from specialists in dances that they won't have learned as part of the standard ballroom and Latin curriculum (off the top of my head, Charleston, Argentine tango, salsa).
How does the public vote relate to the judges' vote? I mean, if the public votes overwhelmingly for a mid-to-low-ranking couple, can that boost them up the board?
Yes. There's a tension between the judges, who score from their professional points of view (although see below!) and the public, some of whom may vote from a position of knowledge but many of whom will vote on likeability, how much they enjoyed the routine etc.
Is the judging always so inconsistent? It can be really weird, yes. And even when judging one couple, sometimes one judge will disagree and give a very different score from another. They all look out for different elements of the dance (technique, mood, showmanship etc) and all tend to have their favourites, even though I don't think they're supposed to.