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Elemental flat damage gets divided evenly between the 3 types (thus you got it right when you said 60 elemental become 20 fire, 20 cold and 20 lightning).
Elemental percentage damage is instead the value you see, applied to all of the 3 elements. Thus, 100% elemental equals to 100% fire, 100% cold and 100% lightning.
Also yes, the sub-element flat damage you get through elemental flat damage also gets buffed by that specific elements +%. Thus, the lightning damage you get through +flat elemental also benefits from +% lightning.
Elemental Damage
Elemental Damage is a combination of the three basic magical elements: Fire, Ice and Lightning. When it appears as flat damage, Elemental Damage will be dealt as 1/3 Fire + 1/3 Ice + 1/3 Lightning, divided equally.
Percent Elemental Damage on the other hand is a bonus to all three elements equal to the full value. It is not split evenly. This makes percent Elemental Damage a powerful way to increase several damage types at once.