Like you are that smart guy, and your best friend just Exalted as a Glorious Bullshit Solar and you have to keep him away from being discovered, stop him from getting the whole power to get to his head and generally make sure he doesn't do something stupid.
Not even the Gloriously Smart Solars could accomplish that, so a vanilla mortal doing so would be breaking my Suspension of Disbelief.
Solars
Solar Exaltations are the most common to appear in crossovers from what I have noticed. I don't know if the same goes for normal fanfiction, but I consider it at least possible. They are powerful, really powerful. To the point where they are gamebreaking in any setting that doesn't have sufficient power-levels. They are also shiny, tend to end up with people that are already awesome or have the potential to be awesome and give them Glorious Solar Bullshit. They are split into various Castes.
Dawns, which are warriors and generals are the first caste of the Solars. I haven't seen much of them actually, they seem to be mostly the swordfighter type. Zenith are priests and leaders, I cannot remember, but I think the Shugo Chara Crossover had Amu as a Zenith, but that is all I can remember. Twilight are magicians, scholars and savants and thus the auto-choice for people most of the time. Honestly, people take Twilight almost as much as they take wizard/sorcerer in a D&D style setting. Nights are assassins, spies and have the whole secret agent thing going for them. I can honestly say that I have never seen anyone use them in a story I noticed. Weird, considering something like a James Bond can easily work. Eclipse are diplomats and I honestly cannot remember where they were used, though I feel like there were one or two cases.
The number of people that could Exalt in crossovers is equal to the number of people that achieve something really awesome. So most protagonists in an action series are good and also a good deal of side characters could work. Honestly, this is probably the easiest choice unless you want to exalt an unimportant side character or someone who never does anything worthy of admiration.
Pretty accurate. They go to people with great heroic destinies that would lack the power to reach them/survive long enough to accomplish it. That doesn't mean they were not awesome in the first place (if their destinies are great enough, it could be reshaping reality to aid them [similar to Domino from X-Men]), but being awesome wasn't enough for Taylor to solo Zion

Dawns can also be Godlike Martial Artists, even being able to learn Sideral Marital Arts, that apply Martial concepts to reality aka "shifting realities weight (like a judo throw)" to inverse gravity. Alexander used a Night Caste as a minor protag in his CG x Exalted fic. In another xover of his Lelouch was an Eclipse (most likely. He ended it early before starting his GSP Lelouch fic who was the Infernal Eclipse Caste, a Fiend). Each of these Caste's also have an inherent Anima power, and each Exalt also has Five Favored abilities, outside of the Five Caste ones they get (these you hinted at, and they are different between Edition 1-2.5 and Edition 3). At most, (assuming the xover is done in the Second Age or later for Exalted), only 150 can be Solars at most (depending on the author since it is canon that the Deathlords can forcibly transform Solar Exaltations into Abyssal Ones).
Agreed on your take on who cannot be Solars.
Lunars
Okay, the Lunars are kind of an odd case. They are partners to Solars and I have honestly not seen much of them in fanfiction. At least in the crossovers I have seen. They just seem to not be important for the most part in these stories or just never appear. I am not really sure how Lunars work, but they have ridiculous shape shifting, are generally crazy good at survival and can generally get almost as good at most stuff as Solars.
They seem to generally be assistants and foils to the Solars, but I really cannot say much more since I honestly haven't gotten into much contact with them at all. I think it could be interesting to see more of them since they appear to be interesting.
Lunars exalt by surviving something that they shouldn't survive by all rights and/or breaking some kind of societal taboo. Some kind of battle would be possible, or going through the wild and eating a human's corpse for survival, or some other thing. An example could be a Misaka Clone in To aru surviving for longer than the original in combat against Accelerator. (I believe Mu-Sensei suggested this at some point.)
I have seen a Lunar Taylor (and Alexander's CG fic had Exalts of all kinds, except Alchemical). In the Second Age, they have generally been creating new races and societies (they have the inherent ability [regardless of Caste] to sire/conceive a unique form of Godblood known as Beastmen, by reproducing with animals [yes, it is possible the ones responsible for Lunars were founders for QQ

). They can also be chosen by making it a key part of their life, protecting something or someone.
Infernals are these that failed, that had the potential to achieve something, but for some reason didn't do it. Maybe they weren't strong enough and lost some big battle; maybe they were scared and gave up or ran away. Whatever, the Yozi send a demon out and give them the chance to gain phenomenal cosmic power in exchange for getting them out of enslavement.
For the most part, this kind of sucks as a way to get someone an Exaltation unless you get creative. It does have quite a bit potential for Anti-Heroes or unwilling villains and you can totally get out of the whole "Free the Yozi" thing if you do it right from what I know. Some potential choices for an Infernal could be a Shirou Emiya who failed, maybe an Arturia? I believe there was a fanfic where Louise failed at the summoning and got a call from Malfeas and there is another with the Baby-Making lady on this website.
Infernals are split between two types, Green Sun Princes and Akuma (I have yet to see an Akuma xover though). Like you said, the Green Sun Princes are people who failed...specifically due to the fact they didn't receive an Exaltation. If the Demon had delivered the Exaltation, they wouldn't have failed. If their were still 300 Solar Exaltations, they likely would have Exalted and come out victorious. The Demons are told to hold it back after finding a suitable heroic mortal, so they fail their hurtle, thus they know, at least in a small way, how the Yozi have felt for 5000 years. Its psychological conditioning and manipulation. Easier to get people to agree to the deal (although they might die from said hurtle, meaning the Demon needs to find another possibility, all the while, the Exaltation is incinerating them from the inside out. If the Demon fails, it will die a permanent death, and even if they succeed, the GSPs have a finite lifespan programmed into the Exalation, and they will die with them; sucks to be a Demon Serf). Getting out of the Reclamation is dangerous, but not as directly hazardous as it is for Abyssals betraying the Neverborn. The only way to truly escape is to become a Primordial (something that, canonically, only Green Sun Princes can accomplish, but is time consuming and during that time, Hell will be hunting you down, while possibly even making deals with the Raksha or Neverborn to hunt you as well, while Heaven is also still trying to kill you [with the burden of the Urge and Torment always remaining {but with an easy Storyteller/GM that is mitigated with Acts of Villainy, although sticking with the spirit of it makes those out of reach, since Acts of Villainy require you to to perform them with the same mindset of their Yozi, meaning that the more you do so, the more you are conditioned to think like the Yozi in question...although in the case of the Acts of Villainy of the Ebon Dragon, that is likely only going to reinforce the betrayal lol}]).
Akuma are humans who effectively "sell their soul" to the Yozi for power. Their bodies, minds, and souls are reforged to make them a kind of artificial Subsouls for the Yozi. They only have a few inherent benefits over GSPs (two I can think of, their Traits can go beyond their normal maximum, and they get a lot of Loaned Experience, with the possibility of trading Negative Mutations for even more experience [without it being loaned]), but Exalted Akuma (those who were Exalted before the deal), can match and even surpass GSPs in several fashions, especially if they were Solars or Abyssals beforehand...no canon Abyssal Akuma to my knowledge though
(Edit: officially, they cannot exist since you can't stack Primordial Essence on Primordial Essence...which makes no sense for two reasons; the GSP are Primordial Essence stacked on Primordial Essence, so said explanation is inherently flawed

and two and more directly relevant, the Abyssals are not using Primordial Essence, but necrotic Essence which if anything is inversed Earth Essence...and Dragonbloods/Solars can be made Akuma, so again, already existing canon contradicts said 'explanation' and wasn't retconned to allow said explanation to make sence)
Baby-Making lady? Mara? Link please
Abyssals
Abyssals are Deathknights who serve the Neverborn. Where the Infernals have it great with good powers and the chance to get freedom, they are kind of screwed. They have to do villainous stuff to not have horrible shit happen to them and generally have to treat others like shit. Except their Lunar mate from what I read. Where Solars make themselves better at things, Abyssals make others worse and seem generally really screwed up.
I think they get chosen from people that are about to die and get the choice for a full heal and an eternal term as deathknights under the Neverborn. They do get awesome (And at the same time horrible) powers from what I understand, but if they don't keep doing horrible shit every so often, horrible shit starts to happen to them and everyone around them.
In addition, if they actually manage to go the redemption route, they don't become free but instead turn into a normal Solar. Which probably means that they lose a bunch of powers and/or turn them into their Solar equivalent. There is no really good Abyssal.
I haven't read anything involving them myself, but they seem rather well liked by people. And I believe I have seen some stuff where the MC is an Abyssal, I think there was something about a Taylor being one? Honestly though, the entire being an Abyssal sucks unless you are already a horrible person and don't mind the mass-murder and perpetual super-villain action
Abyssals also make themselves better at things, just they do so to get better at killing and/or destroying reality, while Solars do it to build, create, and maintain things/reality. Making others worse is more the Ebon Dragon's stick (although Solars and Abyssals can do so in certain circumstances/with certain magic).
The Redemption will refund them all Experience spent on Charms that can be immediately reinvested into Charms. They are suppose to do so along the same route they took as an Abyssal. This route also comes with the benefit of them being free of the Great Curse all Solars suffer, and the Dark Fate Abyssals suffer. Meaning that they get the chance for long term happiness and unrestricted/uncorrupted Glorious Solar Bullshit.
Good Abyssals can exist for a variety of reasons. As you said, they are offered it while they are on the brink of death, by the Ancient Ghosts of former Solar Godkings, who could probably convince Bill Gates to sell all of his property for a glass of water, while he was already holding a bottle of the best water on Earth lol. Simply put, them taking the deal does not mean they were inherently evil, that usually comes with the social reconditioning (read brainwashing) done when they come to the Underworld.
Deathknights need to nigh constantly do evil shit, to keep bad things from happening to themselves. With certain magic, they can kill to reduce their Resonance (and thus escape the consequences of the Dark Fate), otherwise they need to make an extended prayer to the Neverborn to lower it...and if it fails, or if they have a positive intimacy, it will backfire instead.
Terrestrials
Or as I prefer to call them, Dragonblooded, is where it gets interesting. Because I haven't seen a single story with a Dragonblooded protagonist aside from the one short-lived one where Louise summons a freshly Exalted Pseudo-Saito from Creation? I am not sure why, maybe because they aren't as flashy or awesome, or because they don't get as powerful? Though I guess a part of it could be that Dragonborn are well, team players. They are soldiers and warriors and as such are strongest when fighting together with others of their kind.
I can see how that would make any story with a Dragonblooded Protagonist harder since you would have to make some others also Dragonblooded to get the full potential. It could be interesting if you do it with some super-hero team or some Sentai group like the Power Rangers though. Since they get chosen by bloodline, it is a bit more complicated to use them in crossovers. But that is what genetics or magic bullshit is for.
The other problem is that you HAVE to be descended from at least one of the Elemental Dragons to Exalt as a Terrestrial (why they are called the Dragonblooded). No Dragonblood, no chance to Exalt as one. As you said, it can be done, but it likely will be extremely cliched from chapter one (or an exposition dump of legendary proportions

).
I think one major problem is that a big part of their powers is working with the local rules. They are basically asking the Celestial bureaucracy to do something when they warp reality. In another setting, without the loom of fate... honestly my through osmosis gained knowledge of Exalted isn't enough to tell me if they would even be able to use their abilities there.
According to some examples of magical powers in Exalted, it would be possible, but would have a Willpower surcharge that would be applied on each action. Based on the Spectral Keyword for some Abyssal Charms, and a Ghost Charm whose name escapes me (its in Rolls of Glorious Divinity 2). It would be pricy, at least until a Loom of Fate could be constructed for the reality (likely requiring Solar and/or Primordial [and thus Infernals would be a possibility as well] shenanigans)
The only other kind in Editions 1-2.5 is the Alchemical Exalted. Humans reincarnated into Glorious Bullshit Androids. They like the Sidereal, are chosen through Destiny (although for them it is the Design of Autochthon [aka Loom of Fate 2.0]). They are the Chosen of the Primordial Autochthon (whose name I might be misspelling). Alchemical Solutions is a great worm xover fic of this one.
Interesting thread.
PS: While I have now read a lot of core books for 2E, I am far from flawless in my knowledge, especially with Solars and Alchemicals (need to reread the books on the Lunars and Sidereals though), so my comments may be incomplete, however, I believe what I wrote is not incorrect.