Understand, first of all, that you should take my advice with a grain of salt, especially if a PG disagrees. That said, this is the PPC, and you guys have this weird habit of letting me be critical of things, so let's dive in:
In the random prompt, one of the first lines refers to "said quest." It's possible that I simply missed it, but I didn't see any previous mention of a quest and/or mission. Not a huge problem, but I'm somewhat surprised it slipped past you, since it's near the very beginning of the story.
Continuing the discussion of the random prompt in particular, the explanation of the Dragon Shot simultaneously went on longer than I would have expected, and left me more confused than when I went in. You set up the rest of the scene well enough that if you had simply said "He had more pressing matters to attend to, like a grape-themed gunslinger about to unleash a powerful finishing move, the nonlethal Dragon Shot." it would still have conveyed your message without pulling away from the action to explain what's going on.
A more minor point though: the part where Matthew explains to Ajax what a Sengoku Driver is fits alright, as it's before the fight starts, so taking time to explain doesn't pull readers out of the action, and having one Agent be confused about a specific part of a continuum is a generally accepted way to tell the reader what that specific part is. However, since Ajax has been in the PPC for a longer period of time, and the rest of Ajax's actions indicate that he's well traveled from his time as an assassin, AND you later saying that they're one of only a few teams that work in the Kamen Rider canon, shouldn't he already know what a Sengoku Driver is?
Also, you forgot a comma after the interjection Yes at the beginning of the paragraph where you explain what Matthew and Ajax are doing.
One last thing about the random prompt: why refer to the Agents that Matthew and Ajax fight as replacements? The Nora replacement especially seems like it would have been easier to just say she was a hammer-wielding Agent, and indicating that every Nora replacement have a corresponding Ren replacement came across as plain weird. And while it makes sense that the easiest way to justify having a Kamen Rider in the fight would be to make them a former replacement/Sue/background character/whatever, I don't see why you tell us what particular character our grape-blasting-friend was based on after he's already been defeated. If you really wanted to communicate this fact, it probably would have been simpler to either include it as banter between him and Ajax, or have Ajax deduce the fact in a similar manner to how Matthew decided his opponent was a Nora clone. This last one would also serve to further indicate how much knowledge Ajax's experience with the PPC has granted him.
One problem I have with the control prompt is how often you abbreviate things. I understand not wanting to write out Puzzle & Dragons every time you mention it, but spelling it out once or twice before switching to an acronym would be helpful. Similarly, it seems odd that Matthew's internal thoughts/the narrator would abbreviate Ironic Overpower. The fact that Ajax says it normally later on, and that these are the only two instances of it being mentioned, further convinced me that it would be more clear if you simply called it the Ironic Overpower. I'm more okay with using PJO, since it's Ajax and Matthew talking to each other, so if they understand the acronym, it makes sense they would use it.
Speaking of the Ironic Overpower, you seem to imply that it doesn't apply in DI. I know it's the Department of Improbabilities and all, but this seems just a bit too improbable. Actually, after going through a bit while looking at this point, your description of DI leads me to believe you have it confused with one of the Internal Departments. If that isn't the case, I actually have further questions about why you believe DI (the Department of Improbabilities) has offices, and would have only had Matthew reading about Sues, and not actually going into fics.
Another thing from the same general part of the story that strikes me as odd is how you use "speaking of which" to segue into the console going off, but then it takes two lines of dialogue and a whole paragraph between them to actually get to the console making its usual ruckus. Using the segue usually indicates an immediate action based on the segue, so this delayed reaction just feels off.
In the last line of the control prompt, by the way, you forgot an apostrophe in "partners."
The only real critique I can think of for the bios themselves is the format. I think it would have served you much better to avoid the columned approach you used, and to have instead put headings into the text to indicate a change in overarching topic. That, and you misspelled four as for in the explanation of Ajax's powers.
Overall, I noticed that you have a plethora of one or two sentence paragraphs. Sometimes, this is not only acceptable, but puts a helpful emphasis on what's happening, such as during the food fight where, "The rider charged without hesitation." The brevity of this description is a nice parallel to how instinctively and quickly the rider fights, and highlights the action well. Other times though, the sentence doesn't denote an action of enough gravity to warrant its own line. An example where you probably could have combined the sentence with the previous paragraph with ease would be where you describe Matthew's keeping pace with Ajax. Yes, it's a rather long sentence, but it ties in with the event of the last paragraph, and isn't a heart-pounding action line, so it doesn't really need to be isolated. These super short paragraphs often indicate to the reader that you're padding to make the story look longer, which I honestly don't think is what you were going for here, so I don't want you to give off the wrong impression.
One other thing that affects your writing overall that might be seen as an issue is the tone. I, for one, think that your narration style is fun, and its lighthearted nature shows that you do enjoy what you're writing. However, while I like it in general, it is generally discouraged to use the narration as the delivery for your jokes too often, and the more comical style you use might make people especially apprehensive about this. As such, I recommend scaling back a bit, and writing in a slightly more objective fashion. A lot of your parenthetical asides, while amusing, could probably be cut out without affecting the tone or events of the story, and would reduce what some readers might see as forced humor. In addition, there are a few cases where you address the audience directly, which you should either avoid entirely, or use significantly more. Using the narration as a character can be a useful tool for comedy, and that seems to be your primary goal, so I won't say it has no place here, but you should bear in mind that this is not the style that PPC usually uses, which would make potential future crossover missions more difficult, and that this style is much harder to find a good balance for than simply ditching the idea entirely.
My final critique (I promise, I do go over some of the things I liked after this) is something I touched on earlier in specific examples, but I think applies to your writing as a whole: you explain a lot of things, but leave some other things unexplained. I, for one, am not familiar with Kamen Rider or Puzzles & Dragons. Walking into these two stories, I knew very little about either canon. Despite this, I think you explained things far too often, especially in your random prompt. The format for the prompt samples didn't help with this, as you're supposed to have a reasonably brief story, but especially in the middle of a fight scene, it pulls the reader out of the story when you take time to explain what something is or does. Yes, there are cases where you'll have to explain how something works, and sometimes you won't have the ability to deliver that explanation naturally through dialogue, a character's inner thoughts, or some other in-story method, and you'll have to tell the reader in the narration. You should only do this sparingly though. Look at the part of your story where Ajax activates his Talaria. I have no idea what a Talaria is. I assume from the context that it is some kind of magic that either makes Ajax faster so he can track down the rider, or stronger so he can actually fight the rider once he finds the poor sod. Immediately after typing that sentence, I googled what it means (winged sandals, for anyone who made it this far without looking it up themselves), and I was pretty much right. What I'm trying to say is: you've done a good job of establishing what's going on, so when you come to a part that you suspect a reader might not immediately understand, you should ask someone who you're reasonably sure is unfamiliar with the concept (probably a beta reader, possibly just a friend or family member) if they can guess from the context. If they can't you should consider explaining the idea, but you should usually ask first and explain later.
Now, to prove I'm not a totally cynical jerk, here are some things I like about your stories and bios:
A positive I found in the random prompt, was that I like the idea of just footnoting visual references. While I would normally suggest just describing the item, the images you used indicate that it would take more text to describe than would really be worth the effort, so in order to still convey your vision of the scene in a succinct manner, I think you made a good decision.
One thing that I see that I think is very important is that Matthew has a personality. Based on him being a human from World One named Matt and you being a human from World One named Mattman, I'm guessing he's basically you. A mark of a lousy character writer is where the author's avatar in the story is just an every-man, and isn't particularly interesting. There's a fine line between being a drama hog and being bland, and I think you succeeded in finding it. I do want to ask though, while your bio says that Matthew sometimes "might sound like an asshole," I didn't get that impression of him from the writing samples, except for maybe the part where Matthew leaves a note for a hypothetical genderbent Ren to take better care of her husband. Was this due to the prompts you used simply being a bad fit for showing this character trait, or does the bio overstate his rudeness?
Speaking of less than outgoing Agents, let's talk about Ajax. First of all, his powers are understandably complicated, but your comparing them to the more generally understood Pokemon types was a good idea, and I think it let you communicate your idea clearly. Further, while I can see his hydrokinesis and umbrakinesis being very powerful, most of his actual powers seem to be focused more on status effects or helping his team, so he strikes me as a support mage character, which assuages some of my concerns. But Ajax isn't only his powers, which is a point in your favor all on its own. His personality does not immediately remind me of darkness, water, or Hermes, and while it wouldn't be disastrous if he did, it would pigeonhole the character into being the darkness/Hermes guy instead of just being Ajax. And who Ajax actually is seems to be consistent, relatable, and fit with his history. Basically, you knocked it out of the park. He strikes the right chord of being experienced and jaded, but not emotionless, and being stoic and cynical without being spiteful and unlikable. Long story short, good job on Ajax.
On the note of your Agents, they get along well, but not like best friends, which I can see leading to some good stories. I imagine it will take a while for Ajax to fully warm up to Matthew, but I just described Ajax as stoic int he last paragraph, so that was probably going to happen anyway. Plus, exploring how Ajax copes with having a loose cannon (compared to himself and Levy, at least) as a partner in the wake of almost losing his last partner/best friend/leader/potential love-interest will be useful to grow him further from his, as you put it, "rudimentary personality."
In fact, and this isn't really positive or negative feedback, just advice, the one potential issue I see with these two interacting is that Ajax clearly has more going on. That's not to say that you should make Matthew more complicated, just that you should keep in mind that, while you clearly have plans for how Ajax will grow as a character, you should ensure that Matthew has his time in the spotlight as well. You seem to have done a good job of that so far, with the random prompt clearly favoring Ajax (though, that should only be expected when your options are to follow the gamer/student or a god/sorcerer in a fight) and the control prompt letting you show things more from Matthew's perspective.
All in all, you definitely have a few kinks to iron out, but they're not to major, possibly excepting the tone and explanations, so I can only hope I'll be seeing Ajax and Matthew in a full mission soon. Good luck.
