Purple Dragon Knight Review:
Lore: As I mentioned earlier, I think linking a whole subclass to a single dragon type is too narrow of an idea. These should just be "Dragon Knights," with "Purple Dragon Knights" being a specific order of them. Or if they want to differentiate them from the Drakewarden Ranger, they could connect them to only Gemstone Dragons, making up some lore about the Gemstone Dragons forming psionic bonds with humanoid warriors differently from metallic and chromatic dragons.
I do think there is still room for a Banneret/Warlord Fighter subclass and was initially disappointed they didn't even try to fix the original PDK, but also agree that a subclass with "Dragon Knight" in it should have something mechanically to do with dragons.
Knightly Envoy: This ability lets you learn Draconic or some other language and cast Comprehend Languages as a ritual. Intelligence is your spellcasting ability. IMO, Charisma or Constitution would fit better. This is mostly a flavor ability, so there's not much to say.
Purple Dragon Companion: This functions pretty much as the post-Eberron pet subclasses. A nice little buff that I believe came in the new PHB is the change from, "[Pet] acts at the end of your turn" to "[Pet] acts during your turn," so you can have your companion act during your action (especially good for mounted companions). Using Second Wind to resurrect the dragon is good, as Fighters have more than in the original PHB. Compared to the Steel Defender and Primal Companion which require spell slots to bring back, Artificers and Rangers can bring back their companions more if you have multiple fights between short rests that kill the companion, but in a very long and difficult adventuring day the PDK could theoretically resurrect their companion more than a Battle Smith or Beast Master.
The Armor Class scaling off of your Intelligence is interesting. Beast Master animal companions do the same thing but with Wisdom, while Steel Defender's have a base AC of 15 that increases to 17 at level 15. Assuming a Fighter that focuses on Strength or Dexterity (16 or 17) with Constitution as their secondary ability (14-16), a PDK will probably have a 13 or 14 for their Intelligence. So the dragon will probably have a lower AC than a Steel Defender and Beastmaster's companion. And it has the HP of the Beast of the Sky, but with a swim, walking, and flying speed of 30 feet.
It also has darkvision of 60 feet, which is standard. I think giving them blindsight would make sense. Amphibious is situationally useful. It's curious that Stalwart Bond is half as good as Primal Bond. Rend again scales off of Intelligence, so your Dragon will almost definitely be less likely to hit with its attacks than a Steel Defender/Beastmaster companion, and worse than you too. Hell, the bonus to hit doesn't even increase with your Proficiency Bonus, so good luck hitting anything with it at higher levels. Rend's damage also scales off of Intelligence and doesn't even get the base +2 damage that the Beast companions and UA Battle Smiths get.
The best use of Gravity Breath is to pull creatures towards the dragon while it's flying over them. Then they'll take falling damage and fall prone if they fail the saving throw. Otherwise this is very situationally useful, only if there are environmental hazards (cliffs, lava, acid, etc).
Dragon Rider: Pretty similar to the Drake Warden's level 7 ability, in that your dragon grows and you can ride on it. Different in that it can fly with severe limitations while you ride it and you can mount/dismount it more easily. The breath weapon having a larger area and doing a bit of damage is good, making it actually useful to use most of the time. Shared Second Wind is also good.
Rallying Surge: A pretty good ability, a vestige of the "battle commander" theme of the original PDK. But I would prefer something more dragon themed.
Amethyst Pinnacle: Strangely not the pinnacle ability of the subclass. You can finally do the thing you took this subclass to do! Being able to ride on a dragon that can truly fly comes way too late at level 15, which most players will not get to, especially when there are species that grant fly at level 1 (Aarakocra, Fairies, Owlin, etc). I am one of the DMs that doesn't think innate fly speeds at low levels is OP. My Dragon Rider subclass got to do this at level 3.
You also get to replace one of your attacks with one of your dragon's attacks. Which, like, why? Why would you ever do this? If your weapon attacks are somehow weaker than your dragon's Rend at this level, you did something wrong. You will have a higher bonus to hit and deal more damage, can apply weapon masteries, and be able to use magic weapon properties if you have them. This sucks. If Drakewarden Rangers got this ability they
might consider using it, as level 15 Drakes deal 3d6+Wis damage, and have a better attack bonus. This version of a Dragon Knight will use this a single time before realizing it sucks and never doing it again. Being able to replace 2 attacks with Gravity Breath is also probably not worth it outside of rare situations.
Enduring Commander: Terrible name for a boring feature that just gives you and your dragon 2 damage resistances. Wow. Amazing capstone ability. Well done WotC.

Capstone abilities should be powerful and exciting. Giving 2 rare damage resistances is neither of those.
Overall I consider this a poorly designed subclass and would prefer for it to have a broader archetype. The dragon's attack doesn't scale well, it has poor AC and HP compared to most other subclass pets, Gravity Breath is only situationally useful, and there's a good chance you'll have better bonus action options than commanding your Dragon. The Dragon having "Jack of All Trades" that works on saving throws probably won't make much of a difference.
The dragon's attacks and damage should scale better. It should get damage resistance to an appropriate damage type at level 3. If you could ride on a truly flying dragon at level 7 that might make up for the poor AC, HP, saving throws, and damage. Commanding your dragon should share a bonus action with using Second Wind, at the very least. And when you use Action Surge it should be able to take two actions. I think its abilities should scale on your Constitution too, to make the subclass less MAD. Or Intelligence should be the main ability of this subclass.
Oath of the Noble Genie Review:
Lore: Why are they sworn to all of the Genie types? I thought every genie even within the same type competed against every other genie for power, wealth, slaves, etc. It's a bit weird that they're implied to cooperate so much that they share champions with each other. Maybe this is referring to some FR Calimshan lore I'm not familiar with. This is a bit weird, but I'll go along with it.
Elemental Smite: So, you expend a use of Divine Smite and Channel Divinity at the same time to get one of four effects. I didn't like this sort of mechanic with the Heroism Paladin, and I don't like it here. An ability shouldn't require two separate resources to be expended to trigger. This should function when you hit a creature with a weapon attack and expend a use of Channel Divinity, useable with Divine Smite but not reliant on it. But I do like the idea of Paladin subclasses adding to/changing the effects of Divine Smite.
Dao's Crush - Restraining is always good. This is probably the best option.
Djinni's Escape - Even mistier step. Situationally great.
Efreeti's Fury - Also decent. Maybe it should ignore damage resistance?
Marid's Surge - Good, but situationally amazing.
Genie Spells: I can't really see a Paladin using Mirror Image or Phantasmal Force that much, but most of the other options are thematic and pretty useful depending on circumstance.
Genie's Splendor: Oh boy, now someone can compete with Bladesingers in AC. At higher levels with magical shields/armor, this can get insane. Especially with Bladelock multiclassing. I don't think it needs to be nerfed, but man this is an amazing ability. The skill is nice too. The only issue is you have to plan to be a Genie Paladin from level 1 so you make sure to get the right gear and choose the right stats.
Aura of Elemental Shielding: Simply great. Fighting a Dragon? Give everyone in your party that's in the aura a bonus to the saving throw against the breath weapon as well as resistance to the damage type! You can also use this feature to allow the whole party to ignore the Extreme Heat and Extreme Cold environmental effects.
Elemental Rebuke: This is good. I can't remember which, but there's a paladin subclass that gets a pretty similar feature.
Noble Scion: Now this is how you do a capstone! Minor Wish is great!
I love this subclass. It could use a few tweaks, but I think it's the most solid subclass so far. It also can easily be worked into other settings.