Any character can be a "Chaos Sorcerer"

From PHB2, p138:

Chaos Sorcerer
...
Suggested Class Feature: Wild Magic
Suggested Feat: Arcane Spellfury
Suggested Skills: Arcana, Bluff, Endurance, Insight
Suggested At-Will Powers: chaos bolt, storm walk
Suggested Encounter Power: bedeviling burst
Suggested Daily Power: dazzling ray

Notice that all of those start with "suggested"? There is no such thing as a "Chaos Sorcerer" in 4e; it's not a mechanical term, it's just the name for a build that the designers thought was conceptually cohesive.

Remember that 4e explicitly allows players to reflavor their powers. As long as you don't change what the power does, you can describe it however you want. You can absolutely take Dragon Magic and make a "chaos sorcerer" by simply describing your spells as chaos energy (they'll still do whatever damage type they say they do, but they can look like whatever you want).

As for Wild Soul + Burning Spray, if you roll lightning for your Wild Soul and then use Burning Spray, your Burning Spray will ignore 5/tier lightning resistance on your targets. Which probably won't help much unless you've made it deal lightning damage (using a Lightning dagger or the Arcane Admixture feat, for example; not by reflavoring it, since that can't change damage types).

WoTC have released an Unearthed Arcana Wizard subclass called Lore Master. One of the features this subclass gives is:

Spell Secrets

At 2nd level, you master the first in a series of arcane secrets uncovered by your extensive studies.

When you cast a spell with a spell slot and the spell deals acid, cold, fire, force, lightning, necrotic, radiant, or thunder damage, you can substitute that damage type with one other type from that list (you can change only one damage type per casting of a spell). You replace one energy type for another by altering the spell’s formula as you cast it.

When you cast a spell with a spell slot and the spell requires a saving throw, you can change the saving throw from one ability score to another of your choice. Once you change a saving throw in this way, you can’t do so again until you finish a short or long rest.

This seems to cover both effects that you wish your Mage type character to have.

So WoTC appear to be looking at the same sort of concept that you want for your homebrew campaign.

Looking at this trait however it is evident that WoTC, at least, consider the changing of damage type to be not that big of a deal to game balance, but the changing of the type of saving throw to have a potentially greater effect (and thus needed to be restricted).

Taking your fireball example, changing the saving throw from DEX to anything else would eliminate the rogue’s evasion class feature.

In general the saving throws have been sculpted to the type of effect the spell is imposing.

Some salient examples are:

• area of effect spells are largely DEX saves (fireball)
• a spell effecting a targets mental faculties are usually WIS or INT saves (Fear or Feeblemind)

Important Caveat

It obviously comes with the caveat that this subclass hasn’t been playtested as extensively as the subclasses published in the books, and as a result is not necessarily balanced within the game.