This is a great debate! Forgive the tardiness of my two cents.
Whether or not I agree, RAW, a character taking Total Defense can flank, as can one who is flat-footed (without Combat Reflexes or Uncanny Dodge) and/or grappled, as long as he or she has reach >0 ft. and a melee weapon (or Improved Unarmed Strike, claws, etc.). RAW, they are all able to flank because they are not specifically denied the ability to make a melee attack.
In the case of grappling, it’s most clear:
“A grappled creature is restrained by a creature, trap, or effect. Grappled creatures cannot move and take a –4 penalty to Dexterity. A grappled creature takes a –2 penalty on all attack rolls and combat maneuver checks, except those made to grapple or escape a grapple. In addition, grappled creatures can take no action that requires two hands to perform. A grappled character who attempts to cast a spell or use a spell-like ability must make a concentration check (DC 10 + grappler's CMB + spell level), or lose the spell. Grappled creatures cannot make attacks of opportunity.”
A grappled creature cannot make AoO, but can attack if properly armed, and can thus threaten, RAW. While many Conditions disallow most physical actions and thus unambiguously restrict threatening and flanking (cowering, dazed, fascinated, nauseated, panicked, paralyzed, pinned, and stunned) the words “cannot make attacks of opportunity” among all the Pathfinder conditions, are only used for grappled and flat-footed: "A character who has not yet acted during a combat is flat-footed, unable to react normally to the situation. A flat-footed character loses his Dexterity bonus to AC and Combat Maneuver Defense (CMD) (if any) and cannot make attacks of opportunity, unless he has the Combat Reflexes feat or Uncanny Dodge class ability."
Unable to react normally is not unable to act. A GM could rule that before your first turn in the initiative order comes up you are unable to act and thus cannot flank. However, “cannot make attacks of opportunity” is written rather than “cannot act,” and so, like the blinded, the grappled, and the total-defending, RAW, a character not specifically prevented from acting can still flank, if properly armed.
Similarly, attacks of opportunity aren’t among a disabled or staggered character’s specifically allowed actions, but they are not denied the ability to threaten, and so too may provide a flanking bonus. Whether or not they are allowed AoO? That’s for another thread.
Even a blind character, who treats everyone as having total concealment, and is consequently denied AoOs (“You can't execute an attack of opportunity against an opponent with total concealment, even if you know what square or squares the opponent occupies.”), would be able to provide a flanking bonus, since he or she is still allowed to attack.
In fact, let’s have a pair of orcs battle a tag-team of goblins, while all affected by greater invisibility and lined up like so: Orc 1, Goblin 1, Orc 2, Goblin 2
RAW, Goblin 1 and Orc 2 are both flanked, even though no one can see each other.
Quick review of points we’ve made, you all more thoroughly and eloquently than I: (1) RAW, flanking requires threatening but does not specifically require the ability to make AoO. (2) RAW, grappled characters, flat-footed characters and characters using total defense cannot make AoO, but are not specifically restricted from threatening and can find a number of ways to act. (3) “You threaten all squares into which you can make a melee attack, even when it is not your turn.” meaning if you are properly armed and are not prevented from attacking, you threaten (4) Nowhere is it written that running out of AoOs means you no longer threaten. Logic me all you want, but it’s never stated and, IMO, that is not anything a GM was meant to track each round.
So, RAW, a flat-footed fighter with a sword drawn provides a flanking bonus, even without Combat Reflexes, or with Combat Reflexes but after making all his or her AoO for the round. Okay. What about a sleeping monk? RAW, an unconscious character is not specifically denied AoO, nor is a character that is dead, helpless or petrified.
RAW, you could take our sleeping monk, handcuff him, gouge out his eyes, tear off his legs, petrify him, then grapple his rather unfortunate statue. He’ll still give his allies a flanking bonus. 3.5 FAQ says the dead condition means you’re treated like an object, but its not PF official. So, after you’ve finally killed him, make his corpse into a throw rug... just don’t turn your back on it...
RAI, you cannot provide a flanking bonus when dead, helpless, petrified or unconscious. RAI is foggier with blinded, flat-footed, grappled, and total defense, but if a GM rules that a grappled character can flank, I believe he or she must allow flanking under all four circumstances, as they all specifically deny AoO but nothing else.
So your arcane trickster’s pig familiar might have a +4 AC as he flanks your foe. +6 if Babe’s an Acrobat. Any GM that can’t bulldoze that little exploit isn’t really trying.
