So my question is now, suppose I have Obsidian Fireheart (or any other elemental with at least 3 toughness) in play along with two Incandescent Soulstokes, then obviously both my Soulstokes die as they have -1 toughness each, but will Obsidian Fireheart die as well? In other words, when exactly during this process does one copy of Elesh Norn leave the battlefield due to the legendary rule?
The Legendary rule is one of the state-based actions which are performed at several points in a turn.
State-based actions are also what kills created with non-positive toughness.
During the resolving of a spell, state-based actions are not performed. So when the copy of Elesh Norn enters the battlefield, there are temporarily two of them, and your creatures have -4/-4. The 3/3 creatures will temporarily be -1/-1.
Then state-based actions are performed, and those creatures and all but one of those Elesh Norns are killed simultaneously. From that moment, your remaining creatures have -2/-2.
When the second Norn enters the battlefield state based actions are checked and executed at the same time. This means one Born dies because of the legend rule and the 2 Soulstrokes die because of having 0 or lower toughness. This happens at the same time. And your Fireheart lives happily ever after.