More responses!
Arcticblast's The Underdog: As you said, it is quite similar to Zyrefredric's, though more focused. That being said, I do think it suffers from some of the same problems as his. The fact is, we know things like this can work. We see them in the OU meta and can easily point them out. But, of course, we never have gone this route for a CAP project. The problem is, that that itself is not really that much of a concept to follow. "Make a Pokemon that fits in OU but without high BST or wide movepool or whatnot." It's a format, but not really a concept. I almost feel that by being more narrowly focused, you have actually made this vaguer of a concept, since you can't even pinpoint it as being a study of the effects of BST or something. I think you might want to focus more on a certain way of success that does not rely on these things, rather than simply saying that we should make a Pokemon that fits that kind of mold. Though I must say, that fifth question would be a very interesting one to try and answer.
Gloppagus's Weather Lover: The main problem here is that this concept is too specific. Changing forms have major implications not to be taken lightly, and that would predetermine a lot of the Pokemon from the get-go. Not only that, but the entire concept seems near impossible to do without some sort of custom mechanic, which I want to stay clear away from. If you want to do something similar to this though without the form changes, than something revolving around a Pokemon that is actually good in all weather as is would be the way to go. A lot of people are looking at ways to beat weathers, but not so many are looking at ways to make something that helps them.
ChrisTehAwesome's The Glue: This is a cool concept focusing on a role that does not often get a lot of attention. Very few Pokemon can fit that role of being the thing that hold all kinds of teams together. And the few that can are often so generally good that they get banned. I believe that this role has a lot of potential and could certainly teach us alot about team cohesion and teammate interaction. My main worry is on the focus of the concept. To say what I have said many times already, I'd like to see a bit more specifically here on what the goals are to learn about. The questions you have are a decent start, but I'd like to see some more detailed things that can be used as focuses for discussions.
akels's Vola-utility: The main problem with volatile status is that it is unreliable. The ability to simply switch out of it makes it hard to justify using over something that has a more direct effect. Now, obviously, some are better than others. Attraction may last longer, but Confusion is generally superior in that it is easier to inflict and has better consequences when it works. Even so, it usually does little that can't be accomplished greater by other means. The real exceptions to this rule are Taunt and Torment (and maybe Disable). While having a shot of stopping the opponent is generally outclassed, being able to straight up restrict what they do can be quite valuable. As is I don't think the concept has too much going for it, but if you were to refocus on something like these two, there is certainly some potential.
Bashfrog's Lord of Trash: Concepts like this run into problems because they want to make something good, but they are not really sure what or how. Fact is, most lesser used moves are lesser-used for a good reason. I mentioned this in an earlier post with regard to a concept about Imprison. Certain things just cannot shine, no matter how you do it. I'm not saying there are not moves out there that could work for this concept, but I do think you would need to find those moves and focus the concept on them rather than being so general.
Thepoke4ever's Priority King: While slightly different, I do think that this has much the same problems as the other priority concept in that is simply a mold in which we would make a Pokemon, not really a concept detailing what we are setting out to learn. There are definitely things about priority that we probably do not fully understand yet, but we would need more defined goals if we want to get something out of a project revolving around this.
Psylink's Ambush Expert: A lot of what I mentioned regarding the Lord of Trash concept above applies here too. Moves are often bad for a reason, and so I feel that you really need to specify it down to a certain type of move (not a specific one, but just a general idea) if you want this to work. Additionally, the concept as a whole is rather vague. I can't really tell how its use of whatever special move it has is supposed to interact with other sets. It can't be its best option or else it would always use it, but it has to be viable. I feel that this would be very difficult to achieve. I think this really needs more focus on why people would want to use the specific move, and not just more generally good sets.
Unoriginal Name's Luck Counter: Like some other concepets, my main concern here is whether or not this is actually achieveble. There are so many things in Pokemon that run off luck that simply trying to stop it seems rather farfetched. You can easily counter things like Flinch Hax Jirachi with Inner Focus, but there is little out there than can really serve to have major effects on luck as a whole. I don't think the idea itself is bad, but I just can't see it as very plausible to execute. Focusing more on one specific type of luck might be a better way to go about this.
fryfrey's Backstabbing Balance: Well, I'll start of by saying that this kinda of idea has the major flaw of not taking into account the opponents reasoning for not using the CAP. If a Pokemon can out-stall stall, doesn't that mean it is just a great staller and won't it just be used on stall teams? Same thing for any other role. If it can beat something at its own game, then anyone trying that strategy should be using it in the first place. That being said, the last question you have I feel could be the starting point for something very interesting. I don't know exactly how you could work it out, but a study of balance itself could serve to tell us a ton, not just about the current metagame, but about the way we do tiering and the entire game of Pokemon.
WebsterVanCooney's I'm Firein' My Phazers!: Pseudo-Hazing is an important part of competitive Pokemon, and so I do think that exploring it could definitely be worthwhile. Indeed, I have little problem with anything about the general premise. However, as I have said a lot already, I think this is another concept that needs more clear goals as to what we want to be getting out of it. Your third question especially is headed in the right direction as discussing how exactly one gains from phazing, beyond the effects of its namesake move. A little more detail on other points of discussion would be nice to see.
toshimelonhead's Mercenary: Ah, this is a solid idea that works for very much the same reasons as the Perfect Nemesis concept. Now, the obvious difference between the two concepts is that this builds the counter whereas Perfect Nemesis was building the Pokemon to be countered. Both however have the strong points of delving deeper into what exactly makes a Pokemon a counter. Specifically I love the question you have regarding uncounterable Pokemon. Really trying to find if it exists, and if so, what it is could lead to some fantastic discussion.
heartsonfire's Hazard Reversal: While Pwnemon's concept focused more on changing the hazards culture of the game, this concept takes the more direct approach: make life hard for the hazards users while still keeping them as an integral part of the metagame. This is definitely the simpler route to take, but as such, is probably easier to pull off and will likely demonstrate the results of a risky hazards meta better. There isn't really much here that I think could use changing, but I just worry that doing this would be much more restrictive.
Electrolyte's Almost Broken: Oh boy. I remember a concept like this popping up last CAP, and just like then I see a ton of potential here. While short and basic, the first question you present is itself so loaded that it can (and has) spark numerous debates and discussions. Honestly, my two biggest concerns here are the timing and the potential for failure. This concept has a huge amount of potential, but is incredibly risky to do. As great as I think it can be, if screwed up, it could potentially fail like no other. With the CAP project at the point that it is I'm not sure if now is a good time to try something like this. That being said, I will definitely be thinking a lot about this one.
Enguarde's Weather Pressure: One of many weather concepts here, this one is very straightforward in that it wants to be able to simply switch into and threaten the starters themselves. Of all the ways to stop weather teams, killing the starter is obviously one of the most basic. However, the big concern, as always with something like this is that nothing is really stopping it from being used by weather to beat other weather. Also, the general concept always runs the risk of being overpowered if it is going to be able to do what it wants. I'd suggest looking over some of the other weather related concepts and responses to see if you can narrow down a more specific way of dealing with things rather than just trying to beat weather starters.
Scorpio's Slow but Steady: Speed is undeniably the most important part of many a Pokemon, and as you say, is one of the main factors in many of our bans. Exploring what it takes to be successful despite speed could be interesting. That being said, this concept is really way too general. Low speed is interesting, but only in certain roles. You should try and think more about what you want to see out of a low speed Pokemon and what specific kind of role we would learn about with it.
Others1212's Umbrella Head: My concerns here are some of the exact things you say in your post, specifically regarding whether this can actually work or not. I'm also concerned with trying something deemed gimmicky, since that usually translates to not very good. In addition, I just don't think this concept comes across very clearly. Do you want someone who uses weather moves to get rid of the opponents weather, or what? I'd try and clarify more what exactly it is you want this Pokemon to do.
Menace13's The Clutch: Looking at this concept, it almost seems like it is trying to do too much. On the one hand, it is a Pokemon that is better late game than near the start. Ok. Simple enough. But on the other hand you seem to want this Pokemon to have major effects that echo throughout the entire metagame, affecting various playstyles. It really seems to lack focus, so that by the end of reading it, the impression I got was that it would simply be a very good Pokemon. I think you need to focus more on what it means to really be clutch in those lategame scenarios. What exactly should we be looking at to make a Pokemon fill this role?
Little Battler's Tactical Retreat: The ability for a single Pokemon to change from an offensive to a defensive Pokemon is a cool idea, but I can't really see how this would work in practice. Form changing is not really something we want to get into, and most other moves or abilities that seem appropriate focus more on going from defensive to offensive, rather than vice versa. Probably the biggest thing I would like to see here is more of an explanation on how this would even be possible to achieve.
iamdanielcruces's Make Me Bad: While similar in concept to Zyfredric's and Arcticblast's concepts, this one seems to go even farther by seriously saying "make me bad." Well, almost. Like Electrolyte's concept, this could be interesting for exploring the boundary of OU, but instead from the bottom. Just how much do you need to be worth using? What really sets this apart from the rest though is that rather than going "let's be good despite looking bad", this is more like "let's actually be as bad as we can be and still see use." Just like I said to Electrolyte, I really like the potential that this concept has. However, I'm simply not sure how well it would work out. I don't have any real problems with the concept itself, but I'm just not sure if the CAP project itself can work with a concept like this, despite how cool it could be.
And now I'm off to get some dinner. There are a few concepts I have yet to comment on, but I will get to them later tonight. Good work so far people.