Hereof, what happened to jaqen H Ghar?
Jaqen gives her a coin and tells her that if she ever needs to see him again, she must show the coin to a Braavosi and say the words "Valar Morghulis". He tells her to repeat the words and then announces that Jaqen H'ghar is dead. He turns his face away from Arya and when he turns back his face has changed.
Likewise, how was jaqen H Ghar captured? Imo, Jaqen H'ghar was captured because he discovered an extremely dangerous truth, he then trusted the wrong people, was then betrayed, arrested and locked in the black cells. Just so, Jaqen H'ghar is none other than Ned Stark.
Also to know, why did jaqen H Ghar kill himself?
Jaqen killed himself solely for the reason to show to Arya, that he is not Jaqen. He is no one. Every follower of the God of many faces is. They kill themselves and then become someone else, if required, even the person they just killed.
Did jaqen H Ghar want Arya to leave?
It seems like Jaqen never wanted Arya to fail. He constantly gave her slack and even gave her a second chance when she killed Ser Meryn Trant without permssion and then failed to kill Lady Crane as she was instructed to do. Well tonight, she decided that she was Arya Stark and she was going home .