OK. I'm putting together a list of recommendations on this thread. You might already know them in which case my apologies - but if not, here are some recommendations.
I want duels to have a give and take, and for characters to be encouraged to use different fighting styles in different circumstances, like for instance The Chatty Duelists in Princess Bride.
Fate, Mouse Guard, Dogs in the Vineyard. It depends on what exactly you mean.
I wish a role playing game could be highly descriptive without being rules heavy. Not just in combat/dueling - but in all manner of random interactions (npc initial responses, shop contents, you name it).
Fate, Cortex Plus family (Smallville, Leverage, Marvel Heroic Roleplaying, Firefly), Apocalypse World Family (Apocalypse World, Dungeon World, Monsterhearts, Monster of the Week, Saga of the Icelanders, etc.)
I want to play an RPG version of Calvinball.
Wushu Open? Calvinball as high action ridiculous martial arts where you literally can get extra dice (to a scene cap) by declaring you're dodging through a hail of bullets.
I want an RPG that makes exploration/chases/vehicular combat/mass combat more tense and engaging than what I could get by free-forming it.
Apocalypse World and, to a lesser extent, Dungeon World and Monster of the Week. AW is designed on a freeform model where you roll only at points where you'd hand over in freeform RP (and still hand over so the flow is minimally disrupted) - and every single roll adds tension and interest you wouldn't get with pure freeform. (Dungeon World and MotW are of course based on Apocalypse World)
Fractal complexity. Let me either handle a situation in a simple way or apply a more complex, rules-oriented approach, and all with the same characters. Add on the possibility to switch complexity while running a given situation. Let the PCs first wade through simple minion-like enemies with very few rolls and switch to the detailled combat system for all when the boss appears.
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Fate Fractal tries to do this. I don't know if it's what you want but it's worth a look.
I want an rpg that seamlessly and logically extends to mass combat with miniatures and kingdom management.
I believe that Adventurer, Conqueror, King tries to do this.
I wish systems had a way to simulate the 'give up at gun point' instead of calculating that the gun/bow only does 1 hit and it takes multi hits to drop them...
Fate, Smallville, Firefly, and Dogs in the Vineyard all have systems that encourage you to give in rather than take the bullet even if you won't die.
I also want standoff rules, too often it doesn't matter if you are unarmed, you stand a good chance if you just rush a gunman.
Fate, Leverage, and Firefly here. You declare the gun either as an aspect or an asset.
A constructive world generation system which results in descriptions more poetic than prosaic thereby opening up that kind of verbal composition as a part of the player game too.
Have a look at Dresden Files (and its city generation), and Apocalypse World. Also if you're looking for games to bolt onto other games, Microscope or Kingdom are possibles.
I would like the system to have nice incentives (or penalties for not) for PC to purchase stuff other than adventuring gear (a house, for example).
oD&D, Arneson's rules. Where you didn't get the XP from GP unless you spent it on non-adventuring equipment. GURPS social status rules.
... combined with a way to knock people out like in the old crime noir movies so that I could play a Garrett P.I. game.
Fate, Leverage, Firefly are all possibles here.
Give me a game where combat is highly tactical and balanced, but does not involve much math. Bonus points if a major conflict may be resolved in 30min.
Iron Kingdoms RPG might well be what you are after. The wargame came before the new Iron Kingdoms RPG and is highly tactical and balanced and 30mm and I believe cross-compatable (disclaimer: I've played Warmahordes but never the IKRPG).
Give me a game where belief dynamics of big groups of people (political preferences, religions, advertising) are modelled by solid mechanics.
Fate's the best I've got here.
Give me a fantasy or sci-fi game that uses other "races"/species to help explore humanity, instead of treating humans as a boring, "flexible" default.
Monsterhearts. It's all about this - but the PCs are the monsters.
I would like to see a game that handles damage in combat realistically. A person can be killed just as easily by a knife as by a sword, for example - the distance between the two people is what makes the difference. Bigger, heavier, longer, etc. only give a weapon an edge if you are standing a certain distance from the other character.
For this we want to go back to the 80s. Rolemaster, Runequest, GURPS.
I want a game where monsters/NPCs can be defined in 2-3 lines of space so I can run them without looking stuff up or needing more notes.
I think you're looking for Dungeon World. Other games that do this are Fate, Leverage, Apocalypse World, Monsterhearts.
I know I already did one, and it kind of caught on, but here is another I thought of...
I want a game that allows combat styles to have both advantages and disadvantages, with more then a rock/paper/scissors results...
Burning Wheel or Mouseguard I believe do this.