Base Class: Fighter
The Dragoons are a powerful order of warriors who wield polearms and challenge the powerful beasts of the sky. They wear special armor meant to invoke the imagery of dragons, including helmets shaped like a dragon's head, spikes, and wing and scale designs.
Polearm Master
When wielding a glaive, halberd, pike, or quarterstaff, in two hands increase it's damage die to 1d12
Dragon Knight
As long as you are not wearing heavy armor you do not need to get a running start to jump, and At 3rd level your long jump distance equals (10+Str score), and at 7th level (20+Str score), and at 3rd level your high jump distance equals (10+Str modifier), and at 7th level (20+Str modifier),You ignore fall damage up to your high jump distance as long as you are conscious, and do not trigger opportunity attacks when jumping.
Jump Attack
When long Jumping at least 20 feet you may declare you are making a jump attack, if you do you immediately make an attack on an enemy adjacent to the space you finish your jumping movement in (This takes your action as a normal attack would). On a hit, the target takes an extra 1d6 damage. This extra damage increases to 2d6 at 7th level, 3d6 at 10th level, 4d6 at 15th level, and 5d6 at 18th level.
Lightning Spear
Starting at 7th level, glaives, halberds, and pikes gain the thrown property with the range 20/60, and javelins gain +20/+40 increased range.
Dragon's Force
Starting at 10th level when you hit a foe with a Jump Attack they must make a Str saving throw equal to 10+your Str. Modifier + Prof. Modifier or fall prone, and on a hit you deal Str. Modifier + Prof. Modifier damage to that creature, and all creatures adjacent to the target. Once you use this feature, you must finish a short or long rest before you can use it again.
Elite Dragoon
Starting at 15th level you may jump as a bonus action. Large or smaller flying creatures you hit with a thrown weapon plummet to the ground. When wielding a glaive, halberd, pike, or quarterstaff in two hands gain an AC bonus of +2
Master of the Sky
Starting at 18th level your jump attack ignores resistances. Your high jump distance equals (60+Str modifier) when targeting an airborne creature when making a jump attack. Your first attack on an airborne creature in a round deals +10 damage.
The thing that bothers me most, outside of the mentioned Jump mechanic changes, is the lack of paying attention to how DCs are calculated. DCs start with an 8 + some modifier + prof bonus. If you start at 10, there's not much more room to go. Thinking about why it's done this way, +5 strength and +6 prof bonus at level 20 means a 21 DC to your attacks as a dragoon. Now I get it - level 20 monsters should be able to help against that, but on the flip side, D&D 5E was made to flat and make everything more straight forward. And considering that legendary weapons are traditionally a DC 19 save, that should tell you something. If you take your same bonuses and apply them to a standard DC 8 start, you get a DC 19 save for attacks. That's in line with legendary weapons and still gives you a reason to look for them.
I don't know if anyone is still looking at the comments here but one thing I wanted to say is that it might be a good idea to state whether or not the dragoons boosted jump is affected by the jump spell because even without any bonuses at 18th level with master of the sky that would be a 60ft. jump distance multiplied to 180ft. that's just my opinion.
For the level 3 ability, jump attack, does it use your entire action to do it, or can it work with extra attack?
While Polearm Master is just boosting the current most popular build, a flat +1 damage to each swing isn't an unreasonable bonus. I like a lot of what Dragon Knight does, but the clause seems to just grant the Dragoon a free disengage. The only restriction on making a standing long jump is speed, so with a 10+ ft jump distance you can get out of anyone's reach. I'm not saying this is wrong exactly, "ignores opportunity attacks" is not an unreasonable archetype feature, I just want to be clear about the intentions of the feature. Bonus damage and slow fall and free disengage seem a bit overloaded already, and then there's Jump Attack. Is a jump attack a special action that causes you to make a weapon attack? Is it just a modification of the Attack Action? Keep in mind, Fighters have extra attack at 5th and another at level 11. If you're spending you action on something that isn't the Attack action, you're probably dealing less damage that round. I'd suggest cutting Polearm Master (it just straightjackets the build) and reworking Jump Attack as something closer to the Charge trait:
I'm not sure what the point of Lightning Spear is, considering the archetype seems to mostly be about jumping around to reach your targets. It won't really interact with any of the other Dragoon features and doesn't accomplish much you couldn't do already. If you cut the fall damage mitigation from Dragon Knight you could put the monk's Slow Fall here instead, otherwise you should try to come up with something else for 7th.
Dragon's Force makes a little more sense if 'Jump Attack' is a special Action; you're giving up one or two attacks to apply something like the battle master 'Sweeping Attack' maneuver without needing to beat their AC. I really like this dynamic but it makes the Jump attack a messy problem. If you're using my suggestion from above, then you'd just define this special action in the Dragon's Force trait (call it 'Leaping Strike' or something to distinguish it from the 3rd level feature).
Elite Dragoon seems confused about some things. Jumping is already simply part of your movement, there's no benefit in adding a bonus action cost to it. The trick of knocking down fliers is cute, but (as with the 7th & 10th level features) the more substantial benefit is the additional d6 of damage per turn from Jump Attack. This will probably feel underwhelming for some, but I think it's fine.
Master of the Sky feels a bit lazy. Most attackers at this point can get the benefits of a magic weapon against enemies where needed; if a creature has resistance to piercing or slashing damage from magical weapons I'm not sure it should be bypassed so casually. The additional jump height is extreme to the point of parody, but it isn't a problem. The last benefit is somewhat superseded if you use my suggested Jump Attack revision, it'll trigger if you leap 20+ feet up to attack a flying creature. This is another feature where I think you need to go back to your source material (Final Fantasy?) to find a new concept of something compelling here.
Probably the best version of a dragoon I've found for 5e