Frankly, I don't see how these abilities make the job more like a "classic FF Dragoon" considering most classic FF DRGs didn't do much outside of jump and fight with lances, and none of them ever utilized their wyverns/dragons.

So let's see what exactly is overpowered/unnecessary here:

Pet Command: Dragon Flight (Wyvern Stance)
When you use any jump, your wyvern accompanies you into the air and keeps you in flight for up to 15sec, raising you higher and higher the longer you're in the air. When you descend from a higher altitude, you land with double the normal impact force. It is possible to descend sooner than 15sec with the "descend" command.
Well this is already kind of met with Soul Jump, which does critical damage when your wyvern is present. The ability to disappear from battle for 15 seconds is kind of cheap, and can really screw over your other party members in certain situations where they need damage. This a concept that works in other FF games because you're...well, in control of the entire party and really just want to ensure survival. If you need to dodge an attack, that's what Super Jump is for.

Pet Command: Holy Crest (Wyvern Stance)
Duration: 5min
Recast: 1min
Fulfilling its role as a holy dragon, the wyvern takes on a protective role, instead of offensive. The wyvern flies higher into the sky, surrounded by the light of its holy crest (like a white, luminous runic gate). (wings stop flapping, outstretched)
* Requires your wyvern present.
* At this higher position, surrounded by its holy crest, it takes reduced damage.
* Its normal attacks are replaced with a "Regen" effect (AoE around the Dragoon) (at its attack rate, not regen tick rate)
* Every 30sec it can use a "Remove breath" of its own will. (not triggered)
* When the Dragoon uses weaponskills and casts spells, the Wyvern will still use breathes. Elemental Breathes are at reduced power, Healing Breathes are enhanced.
Even if your wyvern isn't doing damaging attacks, having a personal Super Regen/Erase factory with buffed up defense is pretty ridiculous.

Pet Command: Dragon Heart
Level 30
Recast: 3min
Your wyvern can pierce through your enemy and drain its strength for its own.
(glowy transparent wyvern flies through enemy, drawing out red and blue orbs (like drain/aspir) as it returns to your side)
* Requires your wyvern present.
* Recovery of about Dragoon's Vitality x2 for HP and Mind x1 to heal MP.
* Twice as effective against dragons. (undead dragons simply give 0hp and 0mp, no damage)
* If the Wyvern's HP or MP are filled, any excess HP or MP goes to the Dragoon.
* When used as a subjob, this can heal any pet or Fellowship NPC, but the effect vs dragons is dropped.
Kind of unnecessary given that there's Healing Breath and plenty of other ways to use Drain/Aspir.

Pet Command: White Draw
Recast: 5min
Wyvern restores MP to each party member equal to the level of the enemy.
(Makes use of the holy crest in the animation)
* Requires your wyvern present.
* Maximum MP restored to each party member is limited by the highest level of enemy in the game.
* Target does not need to have MP.
Doesn't seem like it'd heal you for much MP. I mean, it's not like I'd turn the ability down, but it's not really all that purposeful. You can use /RDM and use Refresh, get book Refresh, or use Atmas in Abyssea if you prefer the /mage route.

Job Ability: Dragonbreath Comet
Recast: 2 minutes
This jump attack is augmented by the elemental breath of the wyvern. The Dragoon jumps into the air where the Wyvern envelopes his polearm in elemental fire before he plummets toward his target like a comet.
* Requires your wyvern present.
* Must be wielding a polearm.
* Causes magic damage based on elemental breath.
* Damage is increased by Deep Breathing Merits (A new passive effect of having the merits)
As a new jump, this would be too much. We already have two damaging/TP building jumps. A third would be really unbalanced. As an augment to an existing jump, it's not needed. This is essentially combining a jump with Smiting Breath, so..you could just use the jump and the breath by themselves to get the same effect.

Job Trait: Long Reach
Allows melee attacks against "flying" enemies, with polearms. These are enemies that are normally out of reach of normal auto-attacks. (Like Cuelebre, Ouryu, etc)
A flying enemy that can't be damaged by normal attacks is so rare that this wouldn't have use, based on the description. Also it's kind of implausible. Ouryu, for example, flies up pretty damn high. Just having a polearm isn't going to be enough to reach that. But you also explain this:

Long Reach is similar to giving Dragoons a killer effect vs a certain category of monsters, without the intimidation rate. Being able to hit flying enemies is a very unique feature that fits Dragoons. I don't see this as overpowering either. It gives Dragoons the clear upperhand against other melee when facing enemies that like to fly out of melee range where only Ranged attackers and mages could otherwise contribute.
The fact that mages and ranged attackers can attack from a distance while other jobs can't is exactly why this would be unfair if a Dragoon could do it and no other melee class could. Again, it doesn't make sense considering a polearm isn't much longer than 6 or so feet.