Can Morgana play effect (when you play me, deal damage to a unit equals to the damage marked on it)- deal the damage on a Baron with Damage Applied to it via some means? for exampled Firestorm (deal 3 to all enemy units at battlefields)

No — Morgana's play trigger can't damage Baron Nashor.

Morgana, Vindictive's ability ("When you play me, deal damage to a unit equal to the damage marked on it") requires choosing a unit, which makes that unit a target (Rule 355.10; choice made at finalization, Rule 352.4.b). Baron Nashor has "I can't be chosen by enemy spells and abilities" (Rule 756–758), so he's not a legal target for her trigger — same principle as The List vs. Baron (FAQ #11818). Vex-style "programmatic selection" would work, but Morgana's ability is a player choice, not a programmatic selection.

Firestorm can still mark 3 damage on Baron — it only targets a battlefield, not units, so Baron isn't "chosen" there. But that doesn't open him up to Morgana afterward, since her trigger's target check happens independently. And even in a case where a unit could be chosen, Morgana only mirrors the damage already marked — 3 from Firestorm just puts Baron (12 Might) at 6, nowhere near lethal.