Finishing drywall that abuts against metal frame prone to condensation
Searched for interior soffits seems some people use it instead of bulkheads.  From the picture it just looks like you have a door with a header in an exterior wall.  

When you say interior soffit this just means that a portion of the header is air space?  As the thickness of the header is less than the thickness of the door?  Typically you'd insulate the extra space using XPS.  The backside of the drywall that covers the wall above the door looks flush with the interior surface of the door.  Normally you'd use trim instead and have the surface of the drywall flush with the surface of the door frame but using drywall is sometimes done.  In your picture it just looks like they've used the factor edge of the drywall - if that isn't already a j-bead.  I don't think there is much you can do to provide more of a thermal break on the metal frame of that door.  

In terms of moisture getting into the wall above the door, in a cold climate, there should be a vapor barrier (typically 6mil poly) that is on the warm side behind the drywall.  This prevents water vapor from getting into wall spaces and condensing.  Older homes (1920s) used vapor barrier paint.

Since it sounds like you have signs of moisture accumulating in the header space, I'd open it and make sure the vapor barrier is properly sealed with tape or acoustic caulking.  For the intersection between the drywall and the metal door frame I'd be tempted to put sill gasket and then tape poly to the door frame and then put the drywall on top of that and use a slightly wider j-bead to return to the frame and then caulk the jbead to the frame.  If you are already opening up the area around the door it wouldn't hurt to properly air seal the gap.  I used backrod and energy seal and then a passive house tape like tescon vana to bridge between the metal frame of the door and the rough opening wood framing.