No, we sing that for Pischu Li by Hallel.
The mesorah is that Rav Yankel Shtinkavitch made it up in Kletzk in 1874, (some say his brother Getzel made it up)
Nope, I think the middle link is pischu. Top and bottom are the universal yaaleh.
All these niggunim are not chabbad, they are the standard ashkenaz nusach..
Chabbad niggunim are tzoma, 4 bbovas etc. These are totally different styles.