Why didn't the Trace give young Tom away?

I still haven't been able to figure out the "Trace" and, to be honest, I'm not sure J.K. has either.

If I recall correctly, the Trace was first mentioned, perhaps not by name, in The Chamber of Secrets when Harry receives a warning from the Ministry after Dobby's infamous Hover Charm that dropped a big frothy dessert on Aunt Marge. As it developed, the Trace is explained as some kind of charm that attaches (automatically?) to underage witches and wizards to permit the Ministry to "trace" when they perform feats of magic outside school and only lifts once the particular person comes of age.

I wonder, by the way, if the Trace is what permits Hogwarts to locate Muggle-borns with magical powers and then invite them to enroll when they become of school age. But that's an aside...

In The Half-Blood Prince, Dumbledore explains to Harry that the Ministry "can detect Magic, but not the perpetrator" -- that the Trace gives Ministry officials notice only that an act of magic has been performed in the vicinity of an underage witch or wizard. If there are adult witches or wizards nearby, the Ministry simply assumes the magical act was performed by an adult and takes no action. Further, Dumbledore tells Harry that the Ministry cannot tell the difference between House Elf magic and human magic, so Harry took the blame for Dobby's Hover Charm.

A lot is made of the Trace in The Deathly Hallows since Harry, who doesn't turn seventeen until his birthday during the summer before seventh year, cannot start his search for the Horcruxes until the Trace is lifted. After his seventeenth birthday, when Harry, Hermione and Ron encounter Death Eaters in the coffee shop immediately after fleeing the Weasley wedding, Hermione wonders whether perhaps the Voldemort-controlled Ministry has found a way to keep the Trace on Harry even though he is now of age.

Ron objects: it's not possible, he says. The lifting of the Trace at 17 is wizarding law.

So that's the stage against which I ask the following question: in the HBP, why did the Ministry not know immediately, via the Trace, that an underage Tom Riddle had performed the three Avada Kedavra spells that killed his father and grand parents?

Dumbledore explains to Harry that the Ministry did not trace young Tom's magical acts in Morfin's presence because they simply assumed the adult wizard (Morfin) had performed them. But he also tells Harry that "we can be fairly sure what happened. Voldemort Stupefied his uncle, took his wand, and proceed across the valley to 'the big house over the way'. There he murdered the Muggle man who had abandoned his mother, and, for good measure, his Muggle grandparents."

Yet, the Ministry failed to recognise that it was young Tom who killed his Muggle parents, an act that was apparently witnessed by no other person, Muggle or Magical. Should not the Trace have made it very clear to the Ministry that the three Avada Kedavra spells were cast by the only magical person in the vicinity, the under-age Tom Riddle?

Is it possible that Tom forced Morfin to accompany him when he visited the Riddle house to commit the murders, thus leading the Ministry to be more ready to blame Morfin? That would have made the act much more difficult to accomplish, especially without attracting notice from passersby, but it would at least address the potential Trace-related issue.

Another side question arises: why would the Ministry not place a Taboo on the three Unforgiveable Curses so that they have instant notice that the curse has been used and can arrest the culprit immediately? Sure, the Taboo likely only works on words that are spoken out loud (rather than simply thought) but I don't recall a single instance when Avada Kedavra is used, for example, that it is used nonverbally.

The Burrow burnt; the Burrow rebuilt

So I'm watching the last three Harry Potter films... just because. I don't particularly like them but, once in a while, I feel the need to see them again. Especially when I am re-reading the novels.

As I read The Half-Blood Prince (in French), I thought I'd throw in the Blu Ray of the movie version, which led to where I am now, with The Deathly Hallows, Part I, in the Blu Ray player, watching the opening scenes.

And it occurs to me, as it has several times before, to ask the question: if the film-makers decided to take the creative liberty in the sixth movie of introducing an all new scene in which Bellatrix, Greyback and several other Death Eaters attack and finally burn the Burrow to the ground, why is that particular domicile back at the start of the seventh film, rebuilt exactly as it was?

The Burrow was always, in novels and films, a ramshackle collection of rooms, pieced together over years and years, making no sense, comfortable almost in spite of itself. In the books, it never burned so there was no reason for Mr. and Mrs. Weasley to rebuild it into a more practical, more comfortable structure.

In the film, however, we see the Burrow completely engulfed in flames and we get a close up of Molly Weasley, standing by, helpless, as it burns to ashes.

By the time the seventh film starts, however, the Burrow is back in one piece. Exactly the same ramshackle, ill-designed piece that burned only a few months before. Why not improve it? Why not make it a more sturdy structure, bigger, better designed,with better flow, better light, better everything?

Sure, you can tell me that the Weasley's made the decision to rebuild the Burrow exactly as it was before out of a sense of history, of sentiment, of not wanting to lose their "home". But it makes no sense to me.

Is it possible the film-makers simply forgot that they burned it to the ground????