When the combat damage step starts, combat damage is assigned and the 'combat damage' object goes on the stack. Neither player gets prioirty during the combat damage step until after combat damage has been assigned.
So "untapping your stufff during Untap", "declaring Attackers (and Blockers)" and "Combat Damage goes on the stack, I assign damage like so..." are the few times that no one has priority? The combat damage (and the amount of it whether 2 from morph or 6 from Akroma) is set and assigned at this point?
Abilities like Phage's trigger when damage is dealt (combat damage object resolves) and check if the trigger condition is there at that time, so flipping over a Phage while damage is on the stack (flipping over creatures face down due to illusionary mask do use the same timing rules as morph) would result in Phages ability triggering when the damage is dealt (if Phage is in play at that time, of course).
In original situation, the mysterious morph will deal 2 damage, and only 2 damage if both players pass priority, even if Player A flips morph after damage is on stack? (if Akroma, just 2 to face, if Phage, GG)?