Uhh catching up:
I like Jjayy's strategy analysis. Particularly, I agree that Perish Song strategies are only really useful degenerately (part of the reason I ended up heavily nerfing Perish Song in Battle Tree). I do think being able to nullify some of your opponent's options is important in order to make the game interactive and fun, but it should be impossible to nullify all of your opponent's options (assuming the opponent isn't using Magikarp). I suppose that leads me to conclude that when properly-balanced, Perish Song should never work successfully since even allowing a single answer to Perish Song nullifies all its benefits. The only purpose for Perish Song should therefore be as a ploy to drain EN by denying "easy outs" from the opponent and leaving them only their most expensive, resource-consuming answers. I suppose Toxic stall ends up the same way: if a Pokemon can deal the 16+ damage in one hit necessary to defeat Rest, then they probably don't need to rely on Toxic stall at all. In that case, Toxic would be used in situations where you want the opponent to use Rest and maybe Sleep Talk, being very expensive moves, and you hope to set up an opportunity to ENKO.
Anyways, to directly answer the questions:
Underperforming Ability: Gorilla Tactics. +2 BAP in return for extreme lack of flexibility is not good at all. It goes with Choice Band, but Choice Band is bad too. And if Choice Band were a good item, then Gorilla Tactics would still be bad because Choice Band can be equipped to any other mon that lacks Gorilla Tactics.
Well-Balanced Ability: Illusion. Is choosing that one cheating? It's somewhat underpowered, but it has such a unique niche that it runs no risk of overshadowing or being overshadowed by other Abilities. Absolute perfection. I also think Power of Alchemy is extremely well-balanced for similar reasons: a strong identity, a unique effect that gives it room to breathe, and an activation requirement that keeps it somewhat underpowered despite its potential.
Overperforming Ability: None.
Serene Grace, Frisk, Protean/Libero, and Gulp Missile I'd list as particularly above-average Abilities, off the top of my head. I don't think they're overperforming, though, since none of them are so overwhelmingly powerful that they could carry a bad Pokemon.
Underperforming Item: Any berry that activates at an HP threshold. If I have to wait until 50% health to see any benefit from my held item, it better be AMAZING. First of all, I don't even know what Pokemon I'm going to be matched up against when my Pokemon is at 50% HP. We could switch mons and be in an entirely different matchup by then. If I use an item with immediate payoff like RageCandyBar, I get its effect in the matchup I'm currently in, which is a way more reliable way to use my held item than using a delayed berry. Secondly, I'm counting on my berry not being Knock Offed, or Plucked, or Incinerated before I hit the HP threshold, since if that were to happen I wouldn't get any benefit from my held item whatsoever.
Well-Balanced Item: Charcoal, etc. have simple and powerful effects. Only boosting 1 Type of attack makes them more committal then Expert Belt, but they're still remarkably general-use. Having them in BBP as cornerstone aggressive items might level the playing field between mono-type and dual-type Pokemon a little bit, seeing as how now that Rare Candy and Everstone are gone there isn't anything in BBP that explicitly benefits mono-type Pokemon anymore.
Also, Smoke Ball. What a great item. No BAP boost, but allows the user to beat D/E moves. Obviously, it would be better to use a mon that beats D/E moves entirely and give it a BAP-boosting item, but this item provides a useful alternative for mons that would get completely screwed over. Was it well-designed? Maybe not. Seems kinda band-aid design. But I still like the end result.
Overperforming Item: Everyone's already said Expert Belt, which I don't think is game-breaking but at the same time doesn't have any real competition, so I have to agree... But aside from that, Big Root is probably dummy good too, although it's definitely matchup-dependent. And we might want to revisit some of those Signature Z-Crystals that let the Pokemon use a Z-Move twice.
Poorly-Designed Item: Destiny Knot. The effect is great, I like how it basically boils down to "the equipped Pokemon gains a sub-eating move". I haven't found any other items that pressure an opponent's sub-budget so well. What I dislike is the gender requirement. I understand why it's there: Destiny Knot would be unbalanced without the gender requirement, and would go against the flavor of Attract. But the gender requirement just isn't fun to play with because it's too arbitrary. It's un-fun in the same way an item that only works against Pokemon whose names start with vowels would be un-fun. I think gender-based effects should be kept to an irrelevantly low power level because they simply stray too far away from satisfying BBP gameplay tenets.
Well-Designed Item: I have two picks. Firstly, I think the Type Gems (in practice, mostly Flying Gem) are well-designed. Part of the appeal to me is that the Type Gems appear to be strictly "worse" than Charcoal-like items because they're consumable. However, they not only proc consumable-reliant effects like Acrobatics and Unburden, but also can be prevented from activating at will, which can be useful when trying to control your precise damage output. "I might want to equip this item so I can do big damage without accidentally doing too much damage" is such a stupid perk that I love it.
Finally, I think Generic Z-Crystals are well-designed. I like how the obvious benefit of the Z-Crystals (being able to use powerful Z-Moves) sometimes turns out to be less important than the smaller side-effects of equipping Z-Crystals (unable to be Knocked Off, can circumvent Taunt/Imprison/etc). Plus, even though equipping a Z-Crystal requires significant commitment (you can only use 1 Z-Move per game, so equipping multiple Z-Crystals without planning ahead is sub-optimal), there's a lot of diverse options for Z-Moves to use after you've equipped the item. Items with multiple uses and non-obvious benefits can be fun because you get to learn more about them the more you battle with or against them.
I'm starting to think that consumable items give us a lot more design-space than held items.