Salaam Sadder Miyan,
To answer your question about whether the stoning of shiatheen during Rummy is part of Hajj, brother Joseph Islam gives the following answer in one of his articles which touches on the Hajj piligraming from a Quran perspective:
"[with regards to the] Stoning of the Devil (Rami Jamarat). There is no such story of Satan appearing to Prophet Abraham (pbuh) 3 times in the Quran or any other previous scripture. If one visits this location, it is a choice. There is nothing in the Quran to support this practice in which many lives are potentially compromised every year."
Hajj from a Quran perspective:
http://quransmessage.com/articles/hajj%20FM3.htmHassan.