If I reanimate a Body Double, and it copies a Phage the Untouchable, do I lose the game?
Yes--because it comes into play as a copy of Phage, it will trigger as Phage would.
If I reanimate a Body Double, and it copies Norin the Wary, and Norin Triggers, when it (the body double) returns to play to I get to chose a new creature?
Yes; because it has changed zones, it comes into play as a "new" object.
If I copy phage in this case do I lose the game?
Yes--because it comes into play as a copy of Phage, it will trigger as Phage would.
If I reanimate a Body double, and it then phases out, when it phases in do I get to copy a new creature? If I copy phage in this case do I lose the game?
702.23d The phasing event doesn’t actually cause a permanent to change zones or control, even though it’s treated as though it’s not on the battlefield and not under its controller’s control while it’s phased out. Zone-change triggers don’t trigger when a permanent phases in or out.
So you don't get to copy a new creature.