Does an Elder Dragon's effect apply to damage dealt by other units and abilities, or only to damage from the Elder Dragon itself?

Yes — it applies to any damage dealt by Elder Dragon's controller, not just the Dragon's own damage.

"Your damage" means damage marked by you — that is, damage from any deal action you're responsible for, including your units' combat damage, your spells, and your abilities (Rule 142.3.b). Elder Dragon's passive redefines lethal damage for enemy units carrying your damage, so a unit with just 1 point marked by you dies in cleanup regardless of its Might (Rule 142.4.c). The reverse doesn't hold: damage marked by your opponent on your own units is unaffected.

The passive also applies retroactively — damage dealt earlier in the turn is still marked until cleanup, so the moment the Dragon enters, a damaged enemy unit dies before anyone gets priority to react. Preventing the damage (e.g., Counter Strike) is the only out, since prevented damage is never dealt.