Does a GS hive with an Alpha GS only move when one of the Alpha GS icons is activated, or will it also move in an icon of another, "normal" GS, is activated (in the case when that icon is not on the Alpha GS card) ?

As long as the icon on the event card appears on any card in a swarm, the swarm moves/flanks as instucted..

(With the exception of any swarm instructed to flank that already is flanking, which wouldn't move.)