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A class that utilizes high-risk/high-reward mechanics.

Wouldn't that just be taking a penalty to success checks and increasing the effect? For example, PF1E's Power Attack. Seems trivial to implement, but could make a better archetype than a class.

A character who advances the plot by screwing up

I mean, anyone can try and fail and still advance the plot. That's more player choice to choose consequences than character mechanics.

I think the key thing is they somehow screw up, cause serious consequences for others but don't face serious consequences personally. Perhaps the player invokes this ability which results in some kind of bonus to not face the consequence (attacks against them, saves, etc)?

The characters you mention also play off the "lucky halfling" trope. Like the old line "Providence protects idiots, drunkards and children"

the power to manipulate distance

Yeah, I have no idea how to do that mechanically. That would basically need to rewrite all the rules on distance/range/movement. This sounds hard.

steals enemy abilities and uses them

This is a super common archetype in video games. It does kind of break the normal rules about character knowledge of enemy abilities. Quina Quen would be the character I'd base it off of (though Kirby is similar) . It pulls from ancient traditions of eating things to gain their power. Bite off the target's flesh and swallow to understand their powers, another bite and swallow with a declared target power to gain a use of that power. How you get this to progressively scale over 20 levels would be interesting, probably mostly relaxing restrictions as you grow.

I would totally make this Gourmand class. Plus, it gives a setting specific reason why people would want to kill monsters and collect their bodies. Probably can steal some aspects of barbarian and sorcerer. I think there was also a 1e Sorcerer archetype that gained bloodline powers based on drinking the blood of creatures.

I'm making up new classes for my party, and one player has no idea what concept to use for a "not covered by existing classes" character. Power will still be balanced against existing classes, but I will try to make up new mechanics.

Pitch him a character concept that is novel and far outside of what can be done by existing classes.

Include the Chinese ghost (gui), since that's the biggest part of East Asian mythology. There's enough mythology on them to build a series of splat books. Chinese culture has an entire month per year devoted to gui, and the Ghost Festival is a major holiday in many countries.

Some examples:

Egui: hungry ghosts, a key part of Buddhist mythology (see also Preta). Suffering or unfulfilled people who die have their souls remain looking for something. Often it's filling their suffering with food (e.g. when dying from starvation or thirst), but it could be a lust to travel or experience new things they missed in life, wanting to be married or just wanting a proper funeral. Loved ones and descendants can help guide them through their pain to transcendance, with offerings like their favorite foods or burning money or pictures of things they need.

Yaogui: weird ghosts, who practice magic for various transformation goals. Some might be dead gods who want to become living gods again, or humans who want to become demons or demigods, or animal or statue spirits that want to become a living person. This is a huge body of legends which inspired related Japanese myths of yokai and mononoke.

Shuigui: water ghosts, often bound to reside in water (or sometimes take the water with them). They can be seen in reflections, and try to drown the living to trade places with them.

Just make sure you've got a sensitive person picking out the terminology and editing, since gui can also be used as an insult (particularly in racial/ethnic slurs).

We don't generally consider people poisoning rats in their house to be evil, but we do generally consider people punching non-consenting adults to be evil.

The biggest ethical question is "can you control the damage"? With punching, someone can surrender and you can stop punching them. With poison, if they surrender, can you stop the poison? If no, then it could end up killing someone who surrendered, which would be murder and an evil act. That is why canonically paladins were prohibited from using poison in some D&D editions.

Multiclassing seems to be intentionally very painful in 2E, which I feel is a better reason for each class to have more options.

Why shouldn't a Rogue get an Int based racket with the Investigator class features as feat options? Why shouldn't a Sorcerer get the option for Mysteries replacing bloodlines? Why shouldn't a Wizard get the option to trade spellbook for a patron? Why shouldn't a Fighter choose Panache as a Feat option?

Even as a skill monkey, you don't need to be trained in 80% of all skills. You just need to be good at the ones you plan on using a lot. I doubt anyone building the character would waste skill increases on training new skills. You'd have at least 7 skills with 1 from background and 6 from class with 10 Int. But of course no Investigator will only have 10 Int.

Compare this to the other 6 skill classes: Alchemist, Bard and Ranger. Bard is a full caster, Ranger is a martial. Is two skills worth sacrificing full casting or being a martial? I don't think it is, because you've already got the 10 to 12 most useful skills - the next two skills have diminished marginal returns.

Instead, I would give a skill increase at level 1, the ability to become a Master in skills earlier (level 5), and more skill feats.

Investigator was originally between Alchemist and Rogue, and there really isn't much space between them in 2e. I think multiclassing is terrible in 2e, but an Alchemist MC Rogue or Rogue MC Alchemist are each superior to Investigator right now.

Why does an Int based character needs 8 skills trained from the class? If you're getting one skill from your race, one from your background, 8 from the class and 4 from intelligence, that's 14 out of 16 skills you're trained in. [Lore of course is useless because the player and GM will never agree on the scope of the skill.] Performance is useless for non-bards, so that's really 14 out of 15 skills. Plus, other players at the table probably want to be useful at skills sometimes too.

Of course it should be a racket instead of a separate class. But would that sell more books?

My objective is to maximize damage done on the first round of contact to a giant with a flying two-sword charge. I love Mikasa in Attack on Titan, and wanted to see if I could replicate the flying, two swords simultaneous first strike giant killer in Pathfinder for a home game.

Character gen rules: 25 point buy, 9th level, normal wealth by level, no 3rd party materials, no technological items. A CR 9 Frost Giant has 133 HP, so that's the damage I need for first strike kills for this level.

Here's how this build works:
Round 1) Get Tengu Swordmaster into Tiger Trance
Round 2) Get airborne and climb to 60+ feet
Round 3) Charge from above for insane damage, then activate Glide to break fall

Here's I've got so far:
Tengu
- Alternate racial trait: Glide
- ART: Deft Swords
Unchained Rogue 9 (Scout, Swordmaster)
- Archetypes:
- Swordmaster Trances:
- Tiger Trance (Succeed at a combat maneuver check to use Pounce on a charge)
- Dragon Trance (Dragon Style)
- Leopard Trance (Mobility)
- Scout's Charge (Charge makes target flat-footed)
- Skirmisher (After moving 10', attack action does sneak attack damage)
- Sneak attack +5d6
- Finesse Training (Aldori Dueling sword)
- Rogue Talents:
- Surprise Attacks
- Weapon Training
- Combat Trick
- Getaway Artist (to get Fly as a class skill)
- Rogue's Edge: Fly
Ability Scores:
13/22/12/8/12/12
Feats:
- Agile Maneuvers
- Branch Pounce
- Death from Above
- Two-Weapon Fighting
- Improved Two-Weapon Fighting
- Flyby Attack (from Combat Trick)
- Weapon Finesse (from UC Rogue)
- Weapon Focus (Aldori Dueling Sword) (from Weapon Training)
Traits:
- Sword Scion
- Swordlord's Page
Skills:
- Acrobatics
- Fly
- Climb
- Perception
- Profession (Soldier)
- Stealth
- scattered points to get class skill bonuses
Gear:
- 2x +1 aldori dueling sword
- Celestial Armor and/or Boots of Flying
- Belt of Incredible Dexterity +2
- other stuff

So we have four key rolls to make:
1) DC 20 Fly check to climb
2) (The hard part) Combat Maneuver check with +13 to activate Tiger Trance. If this fails, fly more and look for another opportunity next round.
3) +21/+21/+16/+16 against flat-foot attack rolls with ADS (6 BaB + 7 Dex + 1 trait + 1 weapon focus +1 enhancement bonus + 5 Death from Above)
4) 1d8+8+6d6 Branch Pounce+5d6 Sneak Attack/1d8+8 +5d6 Sneak Attack/1d8+4 +5d6 Sneak Attack/1d8+4 +5d6 Sneak Attack

Best case scenario, you hit for 4d8+26d6+24, an average of 133 damage (max of 206!). If you miss the combat maneuver check, you can either climb more or still get one attack on a charge for 1d8+11d6+8 for an average of 51 damage (and a max of 82). Frost giant has flat-footed AC 21, so probability of 4 hits is over 50% if I succeed at the combat maneuver. Which is a big if since his CMD is 29 (25% chance of success), but you can elect not to use the charge that round if you fail your check. The result is she could be circling in the sky for several rounds looking for an opening before striking.

Any advice on how to get a better generic CMB without losing the ability to average over 130 damage? Or how to improve the damage of the first damage? I'm not interested in anything that requires more than one round of attacks or attacks with something other than two swords.