Icing?
 

USA hockey rules apply here, but I'd guess HS would rule similarly.

Team B pulls goalie, team A shoots from behind center red, puck hits the post (or crossbar), and deflects behind the goal line. Actually, I suppose it wouldn't really matter if the goalie were off the ice or not...

Icing? Rule/case reference?

I feel this should be icing as a "shot" that hits iron isn't, technically, a shot.

Yes, it is icing. USA Hockey Rule 624 (a) states the following: When a player of a team shoots, bats with the hand or
stick or deflects the puck from their own half of the ice
completely beyond the goal line of the opposing team, play
shall be stopped and a face-off shall take place at an end
zone face-off spot in the defending zone of the offending
team.
Nowhere in the exceptions does it note the puck passing through the crease or striking the goalpost. The puck has to have "entered the goal as a result of a legal action" for icing to be nullified and the goal to count.