Why Did Otonashi Disappear?

The main reason why he may have disappeared is that he finally realized that he did all he could, and he did save Angel's life, gaining a 'thank you' from her at the end when she passes. He may have stayed behind long enough to realize he needed to leave.

Do Kanade and Otonashi meet again?

The scene fades to white just before the boy makes contact with the girl. This is likely that Otonashi and Kanade meet each other again in the real world as Otonashi was confirmed to have reincarnated back into the real world by the writer.

How did Otonashi die?

Otonashi was killed 3 times in the first 7 minutes of the first episode. The first time was when Tachibana stabs him in the chest with her Hand Sonic, the second was when Noda hits him 100 times with his halberd, and the third was when he fell for the trap that was at the principal's office.