Why was Draco Malfoy chosen to try to destroy Albus Dumbledore?

Maybe it's because Belatrix Lestrange talked to Voldemort about it.  Voldemort decided if he didn't do this, he would kill Draco and his family.

http://inthecapital.com/2012/08/23/draco-malfoy-chooses-towson-over-hogwarts/#ss__24564_1_0__ss

Why did death give the first brother the elder wand?

Death pretended to greet the three brothers.  The first brother said he wanted the most powerful wand ever made.

Why did Draco Malfoy disarm Dumbledore?

He disarmed him because he was told to by Lord Voldemort.