Why didn't the Ministry detect magic being used by Tonks?
They probably did.
We know that the Trace detects magic being used around underage wizards, but that does not mean that the Ministry will swoop in whenever it is activated.
For example, in Harry Potter and the Chamber of Secrets, they sent an owl, which took a little while to arrive;
At first, it looked as though Uncle Vernon would manage to gloss the
whole thing over. (“Just our nephew —very disturbed — meeting
strangers upsets him, so we kept him upstairs…”)
He shooed the shocked Masons back into the dining room, promised Harry
he would flay him to within an inch of his life when the Masons had
left, and handed him a mop. Aunt Petunia dug some ice cream out of the
freezer and Harry, still shaking, started scrubbing the kitchen clean.
Uncle Vernon might still have been able to make his deal — if it
hadn’t been for the owl.
—Harry Potter and the Chamber of Secrets
Indeed, they probably did detect Tonk's use of magic, and sent an owl. By the time it got to Number 4, Harry and his Guard were probably already in the air. Once they got to Grimmauld Place, they could have informed the Ministry that they had picked Harry up that night, and that any magic detected had been performed by adult witches and wizards.
This certainly must have been done before. For example, when Mr. Weasley picked Harry up from the Dursleys' in Goblet of Fire, he had to deflect things that Uncle Vernon was throwing at him, which presumably set off the Trace. Of course, the Ministry might have been less inclined to be understanding after the events of the fourth book....
Lupin did mention that the Ministry was watching the Floo Network:
“Brooms,” said Lupin. “Only way. You’re too young to Apparate, they’ll
be watching the Floo Network, and it’s more than our life’s worth to
set up an unauthorized Portkey.”
—Harry Potter and the Order of the Phoenix
However, underage wizards are certainly allowed to use the Floo Network: Harry did it in Goblet of Fire. The reason they can't use the Floo Network is because the active cooperation of the Ministry is needed to connect it to a Muggle fireplace:
I had your fireplace connected to the Floo Network, you see — just
for an afternoon, you know, so we could get Harry. Muggle fireplaces
aren’t supposed to be connected, strictly speaking — but I’ve got a
useful contact at the Floo Regulation Panel and he fixed it for me.
—Harry Potter and the Goblet of Fire
Presumably the Ministry would have prevented this contact from connecting the fireplaces this time around (since they were "watching" the Floo Network).
Why didn't the Ministry know the location of Grimmauld Place?
Harry wasn't the only underage wizard there. Harry and Hermione are there as well, as are Fred, George, and Ginny. Any of them could activate the Trace.
However, Grimmauld Place is under a Fidelius Charm, one of the most powerful protections around. Given that it is described as making a location "undetectable," and that all of Voldemort's magical skill was unable to locate the Potters when they were thus protected, it seems very reasonable to believe that the Trace could not reveal its location either.
Why didn't the Trace reveal that Harry was at the Burrow?
It did (kind of)!
The Trace certainly would have been activated if anyone used magic around Harry. We know that some within the Ministry knew where he was. After all, Scrimgeour showed up to administer Dumbledore's will:
They all saw it at the same time: a streak of light that came flying
across the yard and onto the table, where it resolved itself into a
bright silver weasel, which stood on its hind legs and spoke with Mr.
Weasley’s voice.
“Minister of Magic coming with me.”
—Harry Potter and the Deathly Hallows
That said, the Death Eaters did not know that Harry was there:
“A mixture; but to all intents and purposes they’re the same thing
now,” said Lupin. “There were about a dozen of them, but they didn’t
know you were there, Harry. Arthur heard a rumor that they tried to
torture your whereabouts out of Scrimgeour before they killed him; if
it’s true, he didn’t give you away.”
—Harry Potter and the Deathly Hallows
The Trace is probably basically ignored when used in a home belonging to adult wizards or witches:
"Dobby,” growled Harry; this injustice still rankled. “So if you’re
underage and you do magic inside an adult witch or wizard’s house, the
Ministry won’t know?”
“They will certainly be unable to tell who performed the magic,” said
Dumbledore, smiling slightly at the look of great indignation on
Harry’s face. “They rely on witch and wizard parents to enforce their
offspring’s obedience while within their walls.”
—Harry Potter and the Half-Blood Prince
This quote makes it probable that the Ministry ignores activations from homes belonging to adults, or which have lots of wizard children: both of which apply to the Burrow. As such, though the Trace might have been activated, the Ministry might not have "processed" it.
In any case, the protective spells may have blocked the Trace as well (so that even if the Death Eaters had people searching through alerts that would normally be ignored, they would not find it). Of course, these protective spells could not be cast on Privet Drive without themselves setting off the Trace, so they could not be used to allow Harry to Apparate.
