Subjectivity first: Yay, you used my gifts! The little Lord Vyce is absolutely adorable, and now I'm wondering what sort of fighting style one would use with a battlespork. It would probably involve lots of thrusting and blocking, since the sharp bits are only on the end, but there's both space between the sharp bits to intercept a weapon and only a small amount of blade that a block could conceivably dull. I've always hated the way that swordfights look in movies, with people constantly slamming the blades of their swords together while they fight. Maybe you can get away with that if your sword was indestructible, or made of plasma like a lightsaber is, but if you're using a metal blade, you're just going to be getting all kinds of chips in it if you smash the sharp part against someone else's sharp part.
Also, in unrelated news, you putting Linkara and Wheatley so close to each other on Linda's favorites list is making me imagine a probably altogether-too-large number of Atop the Fourth Wall/Portal crossovers. My favorites are probably corrupted!Nimue teleporting Linkara down into Comicron One's inner structure and forcing him to portal-gun through the corridors to escape, since it would be the only way to outwit her sensors, and Lord Vyce's living data form taking over a personality core. It's only blue portals from here on out.
Okay, now to what I really came here for: analysis. First, I'm not seeing anything on this "Tabitha". If she is intended as the Atwoods' partner, she should have a section as well, so that we can see what they look like as a team as well as individuals.
When two people are partners, they would normally need to share an RC and Department. There is nothing wrong with having a brother and sister work in different portions of the PPC, but they would need separate partners in order to do that, because Departments are not interchangeable. If Linda got a Bad Slash overflow mission as part of her Floaters job, would Dean simply not be able to go with her? His purview is Mary Sues, and doesn't necessarily overlap with Bad Slash, but if the two Atwoods were partners, they would need to go on all of their missions together. And if Linda just decided to only go on Mary Sue-related missions so that she could be with her brother, wouldn't it be easier to just transfer to the DMS?
Also, if you are planning on submitting a Permission piece soon, you are going to need to get it betaed before submitting it here, to ensure that you keep spelling and capitalization in order. Saying that someone is skilled in "long-range" attacks is very different from saying that he is skilled in "long-raged" attacks, for example, and "the Board" is very different from "the broad", not to mention a few other miscellaneous mistakes. One swapped letter implied that Dean is capable of causing pain at a distance through the power of his anger, and not only does the Word World jump all over things like that, it just doesn't look very good.
You probably are going to need to explain the "a Mary Sue killed their mother with plot holes that sent them to the PPC" angle before getting much farther. Not only is that pretty dark, it raises a few significant questions, "Why did the Sue attack their mother in the first place?" and "How and why did the Sue open a plot hole to the PPC?" being the big ones. In addition, having "a relative was killed by (opponent), so (protagonist) swore vengeance on all (opponent)s" is one of those bits of backstory that will need to affect the story in some way, or it runs the risk of being trivialized, which is a nasty badfic trait. When you get right down to it, the immediate death of a presumably-beloved family member would have a sizable effect on someone's psyche, and you don't want to run the risk of being comparable to the Sues that have their whole species slaughtered in front of them, but then cheerily go off on adventures and romance a main character and are never affected by their horrible tragedy ever again. It's shaky ground, is what I'm trying to say, because following through could easily be too dark for the PPC, but not following through would be trivializing death.
Their personalities are very bare-bones, but seem decent enough so far. You'll need to elaborate on a few more traits before their final form, but what they have now is a good basis to build off of.
Lastly, miscellaneous questions.
Why does Dean have a mini-sewing machine? What does he use it for? Making tiny scarves for mini-Turrets?
Which version of the Pokémon continuum are they from? The games, the anime, or one of the manga?
When did they pick up all of their fandoms and technology? I suppose the former could be explained away by saying that they showed affinity for them during agent training, but as for the latter, I don't recall the Pokémon continuum having smart phones and wireless Internet. I'd almost say that the continuum seems to have the rough technological level of the 1990s, but it has matter-to-energy conversion, giant airships, the ability to restore life from lifeless material, and whatever process was responsible for creating Golurk and Claydol, so that's not right.