Time for me commit mild tomfoolery with my answers.
Which unoptimized abilities give us the widest variety of methods to execute the concept?
I think that abilities that aren't heavily offensively or defensively biased would help us a lot here. Abilities that are purely offensive like Adaptability and those that are purely defensive like Filter obviously restrict us when we have to decide on a role for either CAP30b or CAP30i. Abilities like Cotton Down and Triage aren't necessarily explicitly offensive or defensive I believe would really spark a lot of discussion strictly offensive and defensive abilities would be unable to get.
Using the examples from before, if we choose Adaptability, I can't really see discussion being expanded a lot beyond "this Pokemon can do a lot of damage with STAB moves." It's even worse if we put it on CAP30i because it can't utilize any items to take advantage of that damage (and stacking it with a item that increases the damage of STAB moves too seems a bit lazy). Likewise, if we choose Filter, it's hard to see us talking about more than "this Pokemon can take a lot of damage, including super effective moves." Both examples are obviously exaggerations, but my main point is that discussion would feel pretty limited if those abilities and abilities similar to them.
On the contrary, abilities like Cotton Down and Triage I believe are really great abilities to choose because they potential capabilities both offensively and defensively can really help with discussion and give us more freedom when we have to choose a pathway we want CAP30b and CAP30i to go down. The Speed drops from Cotton Down, for example, provide speed control that can help greatly for offensive pressure or defensive prominence, while Triage granting recovery moves priority could help offensively for revenge killing and defensively for, well, recovery. These two are obviously not the only ability options, but I feel like they do a good job exemplifying my ideas.
Conversely, which abilities are overly restrictive on future stages?
I noticed
flying moose has already mentioned how Berry-related abilities restrict us, though not necessarily in the way I think why they restrict us. Using a more broader way to describe them,
item-related abilities, while not many, I believe restrict us a lot because it choosing one of them I think would really diminish discussion in the later stages, particularly the one after this one. Since abilities like Cheek Pouch and Unburden need items to actually be useful (most of which are Berries, but I feel like saying that they're item-related instead cover more ground and is easier to follow), choosing one of them would obviously mean that we would need to assign it to CAP30b, the forme that isn't forced to hold a certain nonremovable item. The next stage of this process is to choose what forme to assign our chosen ability on and decide on that forme's role, but if we choose an item-related ability, it would basically defeat a lot of the purpose of the stage and make discussion pretty uninteresting. On the contrary, choosing an ability that's not item-related obviously helps to retain the purpose of the stage, but it will (not may, WILL) also spark a lot more discussion for defining a role because it would be open to both formes.
Which unoptimized abilities have the best chance at proving themselves effective in the CAP metagame?
Excuse my very novice understanding of the CAP metagame for this section.
It has already been mentioned before for this, but
Cotton Down has a lot of utility with speed control in such a fast-paced metagame.
Likewise, I can potentially see
Triage seeing some usefulness because again, in a fast-paced metagame, priority is going to be incredibly useful, especially for revenge killing.
I'm surprised that I haven't seen it mentioned so far, but
Long Reach can be very useful in a metagame where contact moves can be punished simply by them being used, whether it be from being chipped down (Rocky Helmet, Iron Barbs, Rough Skin) to getting crippled by status (Static, Flame Body).
Scrappy may also be an interesting approach, specifically helping to take on Dragapult, which is seemingly the best Pokemon in the CAP metagame right now. In general, though, it still will be helpful fighting off other Ghost-types like Aegislash and Pajantom, but it's utility unfortunately ends there and is fairly metagame-dependent.
Abilities like
Mega Launcher,
Pixilate, and
Refrigerate are probably the most interesting abilities offensively because they can provide damage in a more unique way compared to other offensive abilities.
Steelworker also deserves a special mention as an ability that is pretty unique offensively that isn't as restrictive as the ones mentioned, potentially.
Which abilities are under-explored due to flaws in the mons that receive them or a lack of optimization towards the ability? How might those flaws be overcome?
Simply put, the Pokemon that have access to such abilities are simply outclassed by Pokemon already in the CAP metagame. Decidueye has a pretty useful ability in Long Reach for a physical attacker, but it's literally just outclassed by Pajantom as a Ghost-type physical attacker. It can even utilize it's signature move in Spirit Breaker better! Quite embarrassing for the owl. Solid Rock, Prism Armor, and Filter all do the same thing in taking less damage from super effective attacks, but you have no reason to run even the most viable users of the abilities. Rhyperior is outclassed as a Ground-type physical tank by too many Pokemon to list, there are many other Pokemon to use on hyper offense beside Necrozma, and the only time I should be seeing Mr. Mime is when I'm playing Pokemon Unite. A lot of Pokemon in the CAP metagame can perform any role Eldegoss can perform a whole lot better, especially providing speed control. To add insult to injury, its access to a completely better ability in Regenerator makes Cotton Down farther from the light. Triage would be useful on Comfey if it wasn't outclassed easily as a setup sweeper. Even in roles besides setup sweeper, Comfey would rather run Natural Cure instead.
In order to overcome these Pokemon's flaws so that their useful abilities can see the light, they just simply have to be able to compete with the Pokemon outclassing them and provide a reason for them to be used over the Pokemon in the CAP metagame. The easiest example I can give is the Decidueye being outclassed by Pajantom. If Decidueye had a better Speed tier and a more up-to-par Attack stat, then maybe it could see some competition with Pajantom thanks to Long Reach preventing Decidueye from taking unnecessary chip damage or getting statused. I know Spirit Shackle doesn't make contact, but my main point here is that Decidueye would be able to rival Pajantom as a Ghost-type physical attacker, which in turn can potentially make the metagame more diverse.
Do any abilities fit all three above categories? If so, which abilities and why?
I believe that
Cotton Down,
Triage, and
Long Reach fit fairly well with all the categories.
Cotton Down and its ability to provide speed control I think would really expand discussion on roles and won't really restrict at all at any point in the process, in my opinion. It can prove itself in the CAP metagame because, as I said before, its ability to provide speed control would really help in the fast-paced CAP metagame. Speed control via an ability is also pretty unexplored because of how unviable Eldegoss is (and I suppose that relationship applies to Gooey and Goodra as well), so I think it would make a perfect candidate for optimization.
Triage is probably the most controversial ability out of the three, but I still believe it should be up for consideration. Priority recovery moves opens up a lot of unique ways for a Pokemon with access to it to utilize offensively and defensively, and I believe something like that would provide a lot to discuss. Unfortunately, it does restrict us in terms of moves and subsequently typing to give those moves STAB, so that's the main issue with this one. Regardless, Triage can prove itself to useful in the CAP metagame because, as mentioned, its fast-paced gameplay can be control by the priority provided by Triage. Lastly, it is obviously an unexplored ability because its only user, Comfey, isn't viable at all in the CAP metagame, so it would be interesting to see how we can optimize it utilizing Comfey's flaws as a foundation.
Long Reach is definitely my favorite out of these abilities simply due to how unique it can provide utility. It honestly would help create a lot of discussion on what CAP30b or CAP30i can do offensively or defensively without having to take the risks other Pokemon may need to take. Making moves that make contract usually not make contact anymore seems like a mostly offensive quirk, but it can help a lot in other types of utility such as crippling Pokemon with Knock Off and pivoting with U-turn without worrying about much risk against Rocky Helmet users and Pokemon with abilities that can cause status upon contact, like Static Zapdos. I feel like that explained why it may find usefulness in the CAP metagame as well, being able to take less risks in making contact with certain Pokemon in order to damage them or provide utility. It's also an incredibly unexplored ability considering that it's only user in Decidueye isn't viable in the CAP metagame at all. Plus, some of the sets it can run often have it run attacks that don't even make contact at all, which arguably makes Long Reach a very standout pick compared to other abilities that may not see usage in the CAP metagame but have their capabilities known already thanks to them being used in other metagames.