Please bear with me as I have a few questions about Brawlers and one about Ascetic Style
Question 1
At 4th level, once per day a brawler can unleash a devastating attack that can instantly knock a target unconscious. She must announce this intent before making her attack roll. If the brawler hits and the target takes damage from the blow, the target must succeed at a Fortitude saving throw (DC = 10 + 1/2 the brawler’s level + the higher of the brawler’s Strength or Dexterity modifier) or fall unconscious for 1d6 rounds. Each round on its turn, the unconscious target may attempt a new saving throw to end the effect as a full-round action that does not provoke attacks of opportunity. Creatures immune to critical hits or nonlethal damage are immune to this ability. At 10th level, the brawler may use this ability twice per day; at 16th level, she may use it three times per day.
Can knockout be part of a full attack or Brawlers Flurry? Or is it just a single attack
Question 2
What are the penalties for a Brawlers Flurry with a Double Chained Kama, and do those penalties change at reach?
Question 3
Can Ascetic Style apply Knockout, Brawler’s Strike, and Brawler Damage to the Double-Chained Kama?
Question 4
What is the Brawlers capstone? I would have said it was 'Improved Awesome Blow' but
At 20th level, the brawler has reached the highest levels of her art. The brawler’s maneuver training increases by 2 and her dodge bonus to AC improves by 2. This replaces the 20th-level improvement to martial flexibility.
suggests otherwise. If I wanted to instead pick up 'Perfect Body, Flawless Mind' would I lose Improved Awesome Blow or the 20th level improvement of martial flexibility?
Can knockout be part of a full attack or Brawlers Flurry? Or is it just a single attack
It's an Ex ability without a stated action cost, it can't require an action to activate. In this case, the only thign that is functionable is that it works exactly like Stunning Fist (which it's almost guaranteed modeled off), i.e. it's a non-action that affects your next attack roll.
What are the penalties for a Brawlers Flurry with a Double Chained Kama, and do those penalties change at reach?
Brawler's Flurry works like TWF except that you can use the same weapon for both your mainhand and off-hand attacks.
A double-chained Kama can be either a double weapon or a reach weapon. As a reach weapon, it's a single double-handed weapon, which means if you use it like that with Brawler's Flurry, you get a -4 penalty. You could use it as a double weapon, but you'd have to actually use both sides to get the reduced penalty (-2), and thus would have to enchant both sides.Can Ascetic Style apply Knockout, Brawler’s Strike, and Brawler Damage to the Double-Chained Kama?
Knockout works with any type of weapon, RAW even ranged weapons (and spells with attack rolls!) - if it didn't, you'd need Ascetic Form because Ascetic Style only works on passive class features. The Unarmed Strike class feature (damage portion) and Brawler's Strike both count as effects that augment unarmed strikes, and are thus transfered by Ascetic Style.
What is the Brawlers capstone? I would have said it was 'Improved Awesome Blow' but suggests otherwise. If I wanted to instead pick up 'Perfect Body, Flawless Mind' would I lose Improved Awesome Blow or the 20th level improvement of martial flexibility?
Ask your GM. The text in alternate capstones says "In some cases, a capstone specifies what ability it replaces.", but the Brawler one is the only capstone that does so.
One possible interpretation is that Perfect Warrior replaces what it states, while any other alternate capstone replaces Improved Awesome Blow.