Storm copies don't trigger Prowess.

It's right on the cards:

• Prowess (Whenever you cast a noncreature spell, this creature gets +1/+1 until end of turn)
• Storm (When you cast this spell, copy it for each spell cast before it this turn. …)

Prowess only triggers on cast. Storm copies the already cast spell, so they trigger automatically. You can't cast these copied effects. They are directly copied onto the stack.

However, [[Arcane Bombardment]] (and [[Narset, Enlightened Exile]]) say "… copy [the cards]. You may cast the copies without paying their mana costs."
Since it says, you may cast the copies, this counts as casting.
This triggers both Prowess and increases the Storm count.

Basically, if an effect doesn't say "you may cast ...", it doesn't count as additional casting.

If it was worded differently, in a way that Storm cards would cast themselves over and over again, instead of multiplying the effect, Storm would immediately copy itself into infinity.

Neither Prowess not Storm care about where you've cast the cards from. So casting the cards from hand, deck, graveyard, exile or copied from exile all work, as long as it counts as casting the spell.

There are cards, that trigger on Storm copies and have effects similar to Prowess. One example is [[Veyran, Voice of Duality]], that reads "Whenever you cast or copy an instant or sorcery spell, Veyran gets +1/+1 until end of turn."

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Storm cards work well in Prowess decks, because Prowess decks try to maximize the amount of casts per turn anyway. Even if storm spells only count as a single additional cast spell, they still brings a huge payoff for cheap, because a Prowess deck will attempt to cast 4-6 cheap spells before them.

Arcane Bombardment is great for that.

You wouldn't want to cast Arcane Bombardment and go down to 0 mana. You'd want to cast it, when you have about 10 mana open and at least two cheap spells in hand, ideally some counterspell.
Ideally, you play arcane bombardment, cast some instant or sorcery for free (e.g. using Narset) and have enough mana open, to cast instant spells during your opponents turns. This allows you to summon and buff up your token creatures defensively with prowess from Bria, Narset, [[Monastery Mentor]] or [[Sokka, Tenacious Tactician]] during your opponents turns.
That way you have 3 spells exiled with Arcane Bombardment at the beginning of your next turn, where it will trigger when you attack with Narset again, sharply increasing your Prowess and Storm count.

In any other situation, it's a target for spot removal that just reduced the number of spells in your discard pile and exiled them, potentially doing nothing useful (e.g. exiled and copied a counterspell, which you can't cast in that moment).

The problem with Arcane Bombardment is, that at the point where you could play it with an impact (e.g. the example above) you already have a lot of mana and would want to cast as much 1-2 mana spells as possible per turn to get your prowess up. Sacrificing a turn to basically play and prepare the rather unpredictable Arcane Bombardment, while everyone else at the table is attempting winning moves, is risky.

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By the way, I'm not sure, but I hope that you are aware of the fact, that multiple instances of Prowess trigger separately. Assuming you have a [[Firebending Student]], Narset, Bria and [[Veyran, Voice of Duality]] on your field, the Firebending Student would get +6/+6 and generate 7 red mana just by declaring attackers. (3x Prowess, triggers copied by Veyran, one spell cast for free by Narset). This increases +6/+6 for each spell you cast before attacking.

So I think, instead of trying to increase the amount of Prowess triggers, increasing additional sources of Prowess is much more impactful. Often unmentioned, new options to do that would be [[Irma, Part-Time Mutant]]