You've got questions, I've got answers. Let's break this down.
[quote]It has money to match governments and has provided the Chi-Mechframes that the PCs are going to be using, and presumably has enough political influence to operate internationally without causing too many governments to freak out.[/quote]
Yes and yes.

[quote]The Chi-Mechframes that are currently available around the world (?) get their Cores from small-sized Warped and are produced specifically in Japan, where the organization's headquarters are. So going into this RP then, our characters would be totally fresh potential Frame Pilots, rather than having had any sort of training prior to this?[/quote]
Generation 0 (the prototype) was made in Japan. I was leaning towards 'multinational coalition' in terms of prototype creation, so the production logically would've disseminated worldwide. The efficacy to which anyone could create one is up for debate and largely relies on what the nation has access to by this point in time.
The organization in question is located within a skybase of sorts; I used XCOM 2's flying airship as a point of reference here, except multiple times larger. As for the matter of training... 6 months from the start of the RP would have been the maximum time to have gotten used to the Chi-Mechframe's functionality. Anything before that is up to player preference.

[quote]How big does a Warped need to be to be unkillable through conventional means?[/quote]
As a general rule of thumb, the ones around 30 meters in length are usually killable with very heavy investment of resources and firepower. Below that is probably doable by modern military arms, but it would take a fair amount of concentrated firepower to be able to take them down. The really big/durable ones (read: anything bigger than ~40) are usually too resilient to take down without investing something within the realm of "too many missiles to be worth it".

WITH THAT SAID: Warped are [i]weird[/i], and this is not a 'one size fits all' sort of situation. Variance will be variance and all that.

[quote]And yeah, generally speaking, my thought is that the current incursion of Warped means that there's not enough safe land around to feed large populations (unless there's mysterious tech out there to grow food in labs). The high casualties and low chance of success is part of the point of that initiative, while the voluntary nature of this makes it slightly more palatable that the forced conscription that occurred during the early stages of the war. Think of it as something advertised like "offer up your lives for the chance to open up the future for humanity" or some propagandist bullshit like that. I didn't particularly point towards how large the project was or how many volunteers it has though, so it could be as small as necessary to remain a dirty secret.[/quote]
See, I'm not denying the theoretical possibility of the act and never did, just that the knowledge of which and the requisite factors to be able to execute upon it and obtain visible returns (even if out of sheer numbers) is not something that can be hit with a small sample size other than out of coincidence. Of course, a lot of things in life are coincidences, but if they obtained a success from said small sample, I don't doubt that they would expand recruitment for more tests, which would quickly even out the numbers and likely put at least some of those facts out on the table for people to come by, even if they are somewhat hidden initially.

[quote]Could you explain how the scouting process works by the by? Right now I'm reading it as: this international organization has tabs on medical records on a global level while countries are reduced to city-states and when an asymptomatic Echidna victim emerges, they toss money at the country in order to obtain all their background information, then try to snatch up the Frame Pilot before the country in question does (or just tosses more money at the country to buy the potential Frame Pilot off of them)?[/quote]
Yeah, you essentially have the right of it.

[quote]Oh yeah, some more elaboration on what you mean by an 'investment' would be cool. Does it take a shit ton of resources just to keep Echidna victims in quarantine? Or is that just in reference to the idea of a country letting a Frame Pilot go after having made a Chi-Mechframe for them?[/quote]
Just the latter. Making the equipment is expensive, and letting both the Pilot AND the equipment go would likely be a loss that could not be easily recouped.

[quote]And I'm of the belief that this character's personality would work fine with a cast like this, when you already have sadistic loners and suicidal heroes. I'm fine with sanitizing it though~[/quote]
See, I understand the logic behind the character and [i]why[/i] they are the way they are. Their worldviews make [i]sense[/i] given everything that they've experienced, and I'm not denying that fact. With that said, this line stands out to me:
[quote][quote]Xuan-Yu is a bastard, a callous, cruel bastard who masks his inhumanity with an unburdened smile. [/quote][/quote]
There's a difference between that and, say, the aforementioned 'sadistic loner'; while that character does have problematic tendencies, they're not... I don't know, averse to meaningful social interaction? Like... Heck, since I'm reminded of Metal Gear Rising when I look at the character and their backstory, let's use Raiden as an example. Dude's got hangups, sure, and he's definitely more than a bit psychotic at times. But the way he sees the world and responds to things are representative of his mind, and thus allows for the other characters to respond to those in kind. The character you've posited (and I could be wrong) feels like they're not willing to at least put themselves out there and speak candidly. That lack of trust (even on a base level) could very easily be problematic, and given that the characters would have by this point spent months together, being unable to trust someone (with your life or someone else's) is [i]not[/i] a good sign.

I'm not asking for someone who can throw themselves onto the pyre for someone else, obviously (and it's not like we don't have someone like that already), but dialing back from 'callous and cruel bastard who puts up a front' would most at least put the possibility of reliability on the table with respect to the character. Even if they're not necessarily affable (or anything close to it) 24/7, being able to interact with them during downtime without being shut down by insincere niceties would be a marked improvement on all side, IC and OoC. That's my opinion, at least.

I hope that answered everything?