Sadly, that was just an example of very poor writing, given that Harry Kim (at that point) was only a recent graduate from the Academy and a full lieutenant has to ask him what to do at red alert.
Exactly. Especially since it Tom Paris was able to be promoted after being in prison.
In "TWISTED", they weren't on Red Alert. That alarm was never sounded, though it should have. Maybe what was happening to the ship prevented it from going to that alert. I do agree that it showed ineptness on Baxter's part.
Regarding Tom, he likely was a Lt. before he was disgraced from Starfleet, so Janeway probably simply reinstated what rank he was at.