Is this balanced against original Barbarian paths? No. Not at all.
Let's take this level by level:
Level 3: Building Charge
First off, it's unclear as to how the static points are generated. Is it just if there's an enemy in range? Is it if you strike an enemy in range? Regardless of how the points are gained, at level 3, they can have a maximum of 3 points, meaning they can expend either 1 or 2 points to use an action to shock an enemy. This means they'll likely be taking 1d8 or 2d8 lightning damage, which, at earlier levels, seems on par with default barbarian damage.
However, compared with the level 3 abilities from the Player's Handbook, on pages 49 and 50, we run into an issue that will continue to come up throughout this path: the other abilities tend more towards support abilities than damage. The Berserker gains the ability to go into a frenzy and take an attack on a bonus action, but with the harsh penalty of gaining a point of exhaustion at the end of the rage. The Totem barbarian gains either resistance to more damage, more mobility in combat, or an aura that helps nearby allies.
I think at early levels, this ability might not get used; I'm not great at statistics, so if someone has run the math and can tell me I'm wrong, that's fine, but I believe that for a while, this ability will be worse than the barbarian just attacking normally. Once you start reaching higher levels, this starts to be a lot of d8s of damage that the barbarian can put out. And, notably, this is a magical form of damage, which means that the barbarian will be able to effectively hit monsters that are immune to piercing, slashing, and bludgeoning damage. Whether or not this is unbalanced depends largely on the campaign, but it's worth noting.
Level 6: Fly like the Wind
Meh. Flying is incredibly situational. However, the Totem path doesn't get flying until level 14, so this is still weird. Getting flight at level 6 when most classes can't get it until much later is odd.
Level 10: Lightning Strike
This is similar to the Paladin's "Improved Divine Smite" ability, which is gained at level 11, but also does 1d8 damage as opposed to the 1d6 for Lightning Strike. So, compared to a Paladin, this is okay, but again, it's very different from the PHB options. The Totem path gains the commune with nature spell, and the Berserker gains the ability to inflict the frightened condition, both of which I would classify as "support" abilities, rather than damage dealing ones. This is a decent chunk of extra damage, and at level 15, it'll actually outpace the Paladin's Divine Smite.
Level 14: Electric Discharge
Absolutely not. At level 14, this is 7d8 lightning damage to everyone nearby. At level 20, it's 10d8 lightning damage to everyone nearby, in addition to the fact that at level 20, you have unlimited rages. AOE damage is an enormous gamechanger; there's a reason that AOE damage is pretty rare. Having it be once per long rest helps, but it still doesn't bring this on par with the other Barbarian options at level 14. Again, they don't add damage to the class, but rather tactical and supporting abilities. This is the appropriate level for flight, for example, in the PHB.
Finally, regarding intent:
You say your goal is to have the barbarian be able to do damage at range, but I don't think this will work well for that. The Stormy Aura requires lots of targets within 10 feet in order to charge up points, which I wouldn't consider as being at range. Additionally, the point of the Barbarian is to be in melee. That's why they have such high constitution: they're the front line tanky people who stand there and get hit so that the squishy rogues and wizards can toss daggers and spells around without fear.
I think this is a really cool concept, but I think it's weirdly balanced throughout levels. Compared to the original barbarian paths, the extra damage output of this is just too much, especially when the barbarian paths contain primarily support abilities, and the few abilities (like frenzied rage) that grant extra damage come at a high price.
No
Let's take this feature at a time...
Quick build
This is just flatly wrong. This class is a half-caster, they should not prioritize their casting stat. Glancing down below, this class would always want to prioritize Dex--they are restricted to Light Armor no access to the Mage Armor spell and they don't have Cantrips for steady damage output. So they'll want a high Dex for AC, and to make use of Ranged or Finesse weapons--Str is not a good primary for a class that is restricted to Light Armor.
Proficiencies
Light armor only? This pigeonholes this class into a Dex-build if they want anything resembling durability. They are not a dedicated caster--don't even have Cantrips.
Draconic bond.
Basically a Familiar that can attack at this point. However, unlike a Chain Warlock's familiar, the Pseudodragon does not consume one of your attacks in order to act--as-written, this is a complete secondary character.
Spellcasting
You have the spell range of a half-caster, but weirdly scaling efficiency. Without playtesting, it would be hard to determine how this pans out.
That said, your spell list is basically a grab-bag of the most useful spells in the game drawn from every class's list. Balanced fairly well by the fact that they are mostly (all?) Concentration spells
As an aside--you need to rename this feature. Classes that don't follow the standard rules for spellcasting call their magic feature something else--like a Warlock's Pact Magic.
Fighting Style
You get the same full range of fighting style options as a Fighter does. There is basically precedent for this with a Paladin.
ASI
Normal, whatever, moving on.
Draconic Growth
And here is where the balance starts its nose-dive. Assuming you take a Blue or Gold dragon, you just added a CR 3 companion. At a rough ballpark based on experience, a CR 3 monster is roughly the equivalent of a 4th to 5th level character in combat. So, at 5th level, you are basically controlling 2 characters of roughly similar combat effectiveness.
For comparison, the wolf of a 5th level Beast Master ranger has 1 less AC, 1/3 the HP, and does a fair bit less damage while attacking. And, most importantly A ranger must give up an attack of their own to let their wolf attack. And that's not including the dragon's breath weapon.
I would also note that about this point--your 'pet' is lining up to be as good at Persuasion as your party's social lead unless you have someone with Expertise in Social Skills
Improved Bond
Seems handy in a pinch...not a huge deal, moving on.
Draconic Tendencies
Again, not a huge deal...but I would note there's nothing in there about what to do if you already have the Survival and Nature skills
Draconic Growth
So, the wyrmlings usefulness has gradually been tapering off and keeping it alive has gotten hard. That's gone again.
Assuming you went for optimization and took a Gold Dragon...you are now a level 10 player with a CR 10 companion. Again, based on experience, I ballpark that a CR 10 monster is at least on-par with a level 14 PC. Now you have a pet dropping three (very powerful) attacks per round, as well as their breath weapon--and almost certainly has more hp than your party's Barbarian.
Extra attack
So...now your character is up to 5 effective attacks per round? 2 from themselves, 3 from their Dragon.
Riding Master
A fragment of the Mounted Combatant feat...moving on
Greater Bond
I can see this stacking in pretty terrifying ways. It will always work unless aimed at a creature with Evasion.
Increased Draconic Tendencies
Fluff, moving on
Draconic Growth
Ah-hahahahahahahahahahahahahahahaha
Why are you still in a party? You have a CR 17 companion (significantly more powerful than a level 20 PC) that has Legendary Actions and Legendary Resistance. Other classes get a 4th Attack or regenerating class resources--but you? You get a pet legendary monster.
Subclasses
Weapon Master
Level 3
Seems niche and kinda weird..whatever
Level 7
Eldritch Knights don't get this until level 18
Level 18
Stronger than the Mounted Combatant feat, and that's some pretty serious damage...
Spell Master
Level 3
Okay, now you're almost a half-caster. Still really inefficient
Level 7
The lack of a range on this is questionable--otherwise this doesn't seem too out of place
Level 18
Again, pretty potent, but not nearly as broken as your pet flying apocalypse.
Summary
You have a boss monster as a free-acting companion. There is no possible way this is balanced against the core classes unless the storyline gives them some way to acquire creatures of similar power. Which you could do, of course--but that's not what you asked.
Ultimately, the character themselves is quite lackluster--with underpowered spellcasting, no extra attack until level 11, and being stuck in light armor.
However, they have a boss monster as a pet. The way gameplay with this class would go is that it'd be kinda meh until level 5--then you'd be the strongest character in the party by far. That would then taper off as enemies caught up with your dragon, though you're still controlling 2 full-fledged independent characters--til you had another massive surge in power at level 10 that would largely carry through until max level--with another (frankly obscene) power surge at level 20.
Best Answer
There are some problems
The first problem is that the Rage Mage is thematically lacking compared to other Barbarian classes, but also compared to other homebrew you have asked about in the past. Even the name Rage Mage is only made more bland by the fact that 4 out 5 features names include "magic" or "magical" and 3 out of 5 include "rage".
Rage Magic
The fundamental problem with casting spells while raging is concentration spells. The Barbarian class has great incentives to have a high Con modifier and grants proficiency to Con saves, but what really sets it apart from other gishes (such as the Eldritch Knight) in this regard is the Rage.
Rage halves most of the incoming damage, so if the Barbarian can concentrate on a spell while raging, then breaking that concentration will be an order of magnitude more difficult. Moreover, Rage is an amazing buff and every other buff of this caliber requires concentration.
For these reasons, any subclass that lets a Barbarian concentrate on spells while raging is fundamentally unbalanced. It's not the kind of thing that you can balance out by making other features weaker.
The exhaustion per spell levels cast is an interesting attempt to balance spellcasting, but it just incentivises the player to cast their biggest concentration spell and then ignore casting for the rest of the combat, which is boring. Moreover, this can be done as the first action in combat before raging, effectively skipping the first exhaustion.
From a balance perspective I'm not even sure that the exhaustion hits the right mark. If we assume that for a Barbarian a level of exhaustion is worth 1 minute of bonus action attacks, then I don't think most spells hit that mark.
What I find especially weird is that of all spells you don't allow cantrips while raging, when the ability to cast cantrips while raging would not make the Barbarian stronger.
Wilderness Magic
A few ritual spells is not a problematic from a balance perspective. However, stick to the same spellcasting ability modifier for both Wilderness Magic and Rage Magic.
You might be stretching too thin thematically if you go for both "magical dark energy" and "magical association with nature".
Magical Rage
I think this is too weak. This is the level where players start to expect magical weapons, and it becomes almost useless as soon as you obtain one such weapon. Unless the player intends to fight bearhanded like a monk, but that's a weird incentive for a spellcasting subclass.
Also, don't shorthand the description of a feature just because the effect is common and well understood, write it out in full: "... count as magical for the purpose of overcoming resistance and immunity to nonmagical attacks and damage."
Cantrip Surge
Possibly the only exciting feature of this sublass.
Greater Rage Magic
The reduced exhaustion doesn't solve the issues I brought with Rage Magic, whereas the true sight while raging is cool.
Final thoughts
I think the Rage Mage is kind of disappointing and boring. When you look at successful gish classes they all have something that synergises magic with their core kit:
The only thing the Rage Mage gets in this department is Cantrip Surge, but it's not enough by iteself. The Rage Mage desperately needs at least one other feature that synergises spellcasting with the Barbarian's core kit. At the very least, casting a spell ought to extend the duration of Rage like attacking or taking damage do.
There's a diffence between casting magic with obnoxious limitations vs casting magic that synergises: the former is annoying no matter how strong it is, the latter is fun no matter how weak it is. The Rage Mage falls mostly in the former category.
I think you would be more successful if you designed the Rage Mage with the following approach: find features that synergise spellcasting with the Barbarian's core kit, and then figure out how much spellcasting you can fit in, even if it turns out to be only 1/6 Spellcasting, or 1/2 Pact Magic, or just cantrips.