Good questions, all of them.
Hub leaders can see that a post has been reported as a violation, and if it has been adjudicated, just by looking at the list of posts. Details of who reported it, and who acted as moderator on it, are recorded in a separate "manage violations" board, only visible to hub leaders. Even if a post has been deleted, we will see a remnant header in the list of posts, whereas non-hub leaders see no record of the post whatsoever. Hub leaders will know that something has taken place, but all other posters would never know that anything happened at all, unless they read the offending post before we could look after the situation. It just simply disappears, or it's as if it never happened, as far as most readers are concerned, but for one tell-tale sign. If you ever see a post with a subject line that reads Re: blah blah blah, but there is no link taking you back to the prior post in the thread, you'll know that the earlier post has been deleted.
If a post has been flagged, and a hub leader goes to the "manage violations" board to adjudicate the situation, there is a drop-down menu that permits the post to be "green-flagged" (violation ignored), "red-flagged/not deleted" (warning given by email), "red-flagged and deleted" (self-explanatory), or "ban poster". The latter option is subject to automatic review by Agoracom. Hub leaders are only permitted to delete posts by, or ban posters from, the ranks identified as Mailroom, or Treasurer.
Hub leaders cannot see who gives a thumbs up.
There are two mechanisms by which a post can be deleted. The method I described above leaves a complete record, but there is another option that does not record who acted. It is a swifter method, which I believe was created so that the most egregious of posts can be removed instantly. Unfortunately, as I say, it leaves no record viewable by other hub leaders, and any other hub leader can reverse that decision, also anonymously.
Hub leaders cannot access other peoples' PMs. It remains an open question whether Agoracom can view those messages. One might assume that administrative privileges could permit that to happen, so one should be cautious about what one shares with another over an unsecured connection.
Lar