Does Zada, Hedron Grinder work with Soul's Fire and other similar instants?
No, it would not.

Zada:


  Whenever you cast an instant or sorcery spell that targets only Zada, Hedron Grinder, copy that spell for each other creature you control that the spell could target. Each copy targets a different one of those creatures.


Fall of the Hammer:


  Target creature you control deals damage equal to its power to another target creature.


Since there are two instances of the word "target" (and Zada can't be both), Zada is not the only target. 

I suppose Soul's Fire could target Zada herself, so each creature you control would deal damage to itself