Zada, Hedron Grinder allows you to copy spells for all your other creatures, as long as it only targets him. Thus, you will be casting a potentially large number of spells. Does every copy of the spell count towards a storm count?
Unfortunately no. Storm counts how many spells were cast, where Zada just copies spells already on the stack.
When you cast this spell, put a copy of it onto the stack for each other spell that was cast before it this turn. If the spell has any targets, you may choose new targets for any of the copies
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.
Like the other answers, I'm saying no, copies created by Zada, Hedron Grinder don't count towards storm. But I think it's worth adding some explanation of why.
Casting a spell is a specific process you go through. It's not something that just happens any time an object is placed on the stack. Casting involves moving a card (or copy of a card - more on that later) on to the stack, choosing modes and targets, and paying costs. The full process is described in rule 601.
However, Zada's ability copies a spell which was already cast. Therefore, the copy is "pre-cast" for you - in other words, it gets created, on the stack, in an already-cast state. You don't need to (and in fact, you can't) go through the process of casting it again. Since storm only counts the number of times you (or another player) go through the process of casting, copies of spells don't contribute to storm, since they never went through that process.
Naturally, this also applies to the copies created by the storm ability itself. They don't add to the storm count, which is important to know in case someone casts another storm spell later in the same turn.
Although it's not all that common, there are some cases in which you can copy something and still have it contribute to the storm count. Those cases involve effects which have you copy a (nonland) card somewhere other than the stack, and then cast the copy. (Examples; see also the cipher ability.) Here's the key difference: the thing you copy has not been cast, so the copy is created in an uncast state. Once the copy is created, it works just like the original card. Casting it follows the normal process for casting, and so it does add to the storm count.
Thanks to Ivo Beckers for additional examples of casting copies.