To what extent should this Pokemon be addressing threats? Should it be able to wall everything they use, or should it just be able to take hits well enough to be able to force them out with an attack or status?
For this, I agree with what reachzero had originally posted. We should be able to custom-tailor CAP10 to "Soft Counter" any threat that we see to our team. One thing to keep in mind is that we won't be going around mindlessly countering things for the sake of it. The way I see this being used is as a final pokemon to plug any weaknesses you foresee in your team. It can indeed go beyond that (maybe used for mindgames if u wish), but its main purpose will be to address those specific threats to your specific teams. I think it is enough that this pokemon takes a hit well enough and forces its target out (sort of like how Swampert treats Jirachi).
Should this Pokemon be addressing threats offensively or defensively? Could this also be customized per threat by the user?
I think it should be able to accomplish both, but depending on the Pokemon. I can see it working out both ways here. Let's take the example of Salamence. CAP10 can switch into any of its moves with ease, take a hit (and still have enough to survive more), and threaten it immediately with an OHKO. But it can also work out that it switches in, takes a hit, and threatens to completely cripple it. It should have the option to do both, up to the choice of the user. Obviously it won't be able to accomplish this for every threat, but I don't see the option as being overpowering.
What kind of stats will facilitate a Pokemon that can be tailored to counter any threat? Should it be a primarily defensive Pokemon, offensive or balanced between the two?
I like the previously mentioned idea that we could just take one of the Fairies and tweak them. I envision this as being primarily defensive with passable attacking stats. CAP10 should have a fair amount of bulk, or it simply cannot switch in otherwise. For offense (stats+movepool together), it should have just enough to seriously threaten its target. I'm sort of undecided on whether to give it sub-par attacking stats and a decent movepool or give it high attacking stats with a shitty movepool. I'm slightly leaning towards the former, however.
Just how limited should the Pokemon be in regard to dealing with threats that it isn’t specifically designed to counter? If the opponent doesn’t carry the threat that a particular CAP10 was customized to counter, will it become dead weight during that battle?
It shouldn't become so specialized that it's absolutely useless if your opponent doesn't carry its target. It should be able to at least stand up to a few things (maybe not threaten them). In fact, if you can switch it into something else first and act as if it's a counter, you could end up surprising your opponent when the real target comes into play. If it's totally useless against everything that it's not meant to handle, then CAP 10 is really not worth making IMO. I also feel that it's good enough that, EVEN IF CAP10 can handle more than one threat effectively, tailoring it to one thing will open out some glaring weaknesses that other pokemon can exploit. That is, we don't have to be too Draconian regarding how little it should cover.
-----------
There are certainly a few things in regards to ability and movepool that work here. One set of moves that holds a lot of merit is the bunch of priority moves. Ice shard, bullet punch, and mach punch all deal with some of the premier threats beautifully. They achieve exactly what we want: force the opponent to switch in fear of death. Of course, we do need to prevent it from running multiples of these at once, but it's still a possibility.
For ability, I really like both schools of thought: Multitype AND immunity-based. The main argument against Multitype (strangely enough) is that people seem to find it too weak. Granted, against Salamence specifically, it will not offer you much, and the loss of item usage and ability is a bummer. But it still offers a lot of advantage against some of the other big threats out there. Mono-ghost shuts down most of Lucario's arsenal (barring crunch), Mono-fighting takes out both of Ttar's STAB's, and Specially defensive Mono-steel will comprehensively shut down every variant of Latias (including trick!). There are many, many such options open. And Multitype has enough limitations that it leaves CAP10 with a bunch of glaring weakness to exploit.
That's not to say that good type combo + immunity ability is not an equally viable route. It trades away a degree of unpredictability for item versatility and better staying power. Coverage can be left up to the movepool, and it also opens up the option to use type-resisting berries and other handy items.