1. Who was the only know witch or wizard living in Little Whinging? Harry. Ever since Voldemort's return Harry and Dumbledore had been spreading the word that he was back. The Ministry of Magic wanted everyone to think they were safe, that Voldemort wasn't back, so they tried smearing the two who were letting everyone know the truth.
But for Umbridge it was more than merely smearing Harry and Dumbledore's names, it was about maintaining power. Pushing Dumbledore aside would be pretty easy, he'd turned down the job as Minister of Magic on numerous occassions. And he did not witness Voldemort's return. So it could be claimed that he was telling everyone that Voldemort was back as a means to gain power, to become the Minister of Magic by making others think he could protect them better than Fudge and the Ministry.
With Harry, however, things were different. Still underaged he couldn't be after Fudge's job. But he did witness Voldemort's return which made him very dangerous as his testimony could topple Fudge. So in order to maintain her position as the second most important person in Fudge's regime, she sent the dementors to silence Harry for good.
But of course Fudge's regime fell, not because of anything Harry or Dumbledore said but because Fudge and others who worked at the Ministry saw Voldemort had returned. And somehow Umbridge managed to survive the fall.
2. Voldemort was more afraid of letting Harry see into his mind, and afraid to try be in Harry's mind again. He'd been in his mind at the end of book five, it wasn't just Dumbledore who defeated him that night. Though he had Harry's blood in his veins and could thusly touch him, being in his mind was agony for him.
Being in Harry's mind also meant he'd have to open his mind to Harry, which he was afraid would mean Harry could also use to know where he was. Unfortunately he was loosing his own control as he got upset or excited, letting Harry see into his mind and thus giving Harry the needed edge.
3. The Ministry can't detect where people apparate or disapparate to or from. However, they can detect magic used in the presence of an underaged witch or wizard, which includes apparation. The Death Eaters and their people (folks under the imperius curse) in the Ministry could have detected Harry apparating via side-along-apparation at the start of the book because he was underaged and still had the tag on him. But once he turned 17 he became of age and the tag vanished, which then made it impossible for him to be traced when he apparated.