Why do people disappear? Why do people disappear if they become Angel's friend? Also, why did Otonashi say "we will not disappear, be our friend" (something like that)? 

Why did other Angel's "friends" disappear and Otonashi and others didn't?

("Angel" is Kanade)
Haven't watched Angel Beats! for a while but I tried to do the best research I could.

At the ending of episode 13, it is heavily implied that Otonashi


   did pass on. He reincarnates and meets the reincarnated Kanade.


Regarding people disappearing,


   Yui thoroughly explains that they disappear when they have their wishes fulfilled. They are in "purgatory" because they were unable to experience a normal youth before dying.


However, Angel Beats: Another Epilogue could falsify this, depending on how you view it:


   Otonashi has become the new student council president and is helping the students pass on. With this (OVA?), two endings are presented: Otonashi remains in "purgatory" to wait for Kanade like The Programmer (as the reincarnation has not been described to be limited it is possible that he could wait for many hundreds of years for her) or is just an add-on to the original ending, where Otonashi does move on.