In searching for answers to this, I've come across an interesting point on the Paizo forums. If you are in one of the far corners and you consider these not threatened, then you can diagonally move towards the character and never provoke an AOO (that is, per RAW)... and yet, logically, a threatened area should make an uninterrupted circle around the creature. This may explain why 3.5 made an exception out of this.
Yes, that is precisely why 3.5e made that exception, and it’s also why Paizo issued an official FAQ that changed Pathfinder’s rules to add the same exception, as @caps reports in this fine answer that you should go upvote.
Thus, a reach weapon can attack the following \$X\$’s from \$C\$:
\begin{array}{c|c}
\phantom{X} & \phantom{X} & \phantom{X} & \phantom{X} & \phantom{X} & \phantom{X} & \phantom{X} \\ \hline
& X & X & X & X & X & \\ \hline
& X & & & & X & \\ \hline
& X & & C & & X & \\ \hline
& X & & & & X & \\ \hline
& X & X & X & X & X & \\ \hline
\\
\end{array}
Before the FAQ change
However, the FAQ entry that caps reports did not exist at the time this question was asked. At that time, you did not get the four corners, and so could only attack these \$X\$’s from \$C\$:
\begin{array}{c|c}
\phantom{X} & \phantom{X} & \phantom{X} & \phantom{X} & \phantom{X} & \phantom{X} & \phantom{X} \\ \hline
& A & X & X & X & & \\ \hline
& X & B & & & X & \\ \hline
& X & & C & & X & \\ \hline
& X & & & & X & \\ \hline
& & X & X & X & & \\ \hline
\\
\end{array}
Here, it would appear that someone could step from \$A\$ to \$B\$ to avoid an attack of opportunity altogether. However, even before the FAQ just changed this to be like 3.5e, the developers at Paizo had ...for lack of a better word, we’ll call it a clarification, though it honestly just confused me more. From an earlier FAQ:
Can you or can you not attack diagonally at a distance of 2x squares (15'=10' exception) with a reach weapon?
James Jacobs: Nope. A reach weapon gives a specific extension to your reach. When you count out squares, since every other square is doubled when you count diagonally, that means that there’ll be corners where you can’t reach.
Sean K. Reynolds: It's an artifact of the grid. The closest the rules come to addressing this is in Large, Huge, Gargantuan, and Colossal Creatures, which says:
Unlike when someone uses a reach weapon, a creature with greater than normal natural reach (more than 5 feet) still threatens squares adjacent to it. A creature with greater than normal natural reach usually gets an attack of opportunity against you if you approach it, because you must enter and move within the range of its reach before you can attack it.
So just because the grid has a square for "15 feet away" and a square for "5 feet away," but no square for "10 feet away," using that corner path doesn't mean you're magically teleporting from 15 feet to 5 feet; you are passing through a 10-foot-radius band around the creature, and therefore you provoke an AOO.
Admittedly it's not clear, and obviously it doesn't have the diagram in the 3E book to provide a non-textual example, but it's supposed to work as I described above.
Basically, the idea was, under the rules at the time, you didn’t threaten 15 ft. away, so you don’t get the corner, but you did threaten 10 ft. away and there’s no way to move from 15 ft. away to 5 ft. away without passing through a point that is 10 ft. away. Thus, someone moving from 15 ft. away on the diagonal to 5 ft. away on the same diagonal was going to provoke even under these rules.
So the enemy at \$A\$ moving to the point marked \$B\$ towards \$C\$ with a reach weapon provoked an attack of opportunity (assuming this isn’t a 5 ft. step of course), because somewhere between \$A\$ and \$B\$, there is a point that is 10 ft. away from \$C\$ that the enemy has to pass through.
Presumably, you would have adjudicated the enemy’s position for the purposes of the attack of opportunity as being \$A\$, though this was never made clear. In this sense, the end result was identical to the 3.5e/post-FAQ version for movement towards you: creatures leaving that corner square to enter a square inside your reach provoked an attack of opportunity as if you threatened that square. You were not eligible to make an attack of opportunity if the enemy performs any other action that provokes from \$A\$, including movement in other directions, because you do not actually threaten it.
This was a headache. Even before the FAQ changed things to match 3.5e, that was precisely what I recommended:
Reach weapons are one of the few fairly-nice things that melee can get. There’s really no need to nerf them. I strongly suggest that you straight-up ignore this nonsense and use the 3.5 rule. The exception to the usual calculation of ranges in the case of reach weapons is weird, but clearly there was a good reason for it: without it, you wind up with this mess.
No
…but it's probably fine if you let them.
From a strict reading, bolas can be chucked at an enemy like an overly-elaborate rock to do 1d4 damage, or can be used to make a ranged trip attack.
Since the wording just says "you can use this weapon to make a ranged trip attack", nothing suggests it should operate like the Trip(ex) ability and trigger a trip off a normal attack.
This also matches how e.g. a spiked chain is described: it has a listed damage, and can be used for tripping, but these facts are not directly related.
I would tend toward running it as written. Exotic weapon proficiencies tend to be a high cost for a low return though; if you wanted to soften that cost it might be reasonable to say bolas have a ranged version of Trip (Ex) instead of their ability as written.
On how a "ranged trip attack" works: good question. The trip attack section on tripping with a weapon says "In this case, you make a melee touch attack with the weapon…." However, I would read "ranged" as the more specific rule. Since "make a ranged touch attack with the weapon" parses without any particular difficulty, I would go with that. From there, I would apply tripping exactly as written:
If your attack succeeds, make a Strength check opposed by the defender’s Dexterity or Strength check (whichever ability score has the higher modifier). A combatant gets a +4 bonus for every size category he is larger than Medium or a -4 penalty for every size category he is smaller than Medium. The defender gets a +4 bonus on his check if he has more than two legs or is otherwise more stable than a normal humanoid. If you win, you trip the defender.
Best Answer
Scenario 1: "When my allies are flanking a foe, if I threaten that foe then I count as flanking that foe"
First, you can (but don't have to) remove either one (but not both) of your allies from the diagram and still be counted as flanking that foe by employing the following feats:
The style feat Despana School (Drow of the Underdark 56), in addition to other benefits, allows you to remove either ally from the diagram and still leave you and the other ally flanking the foe if that other ally is a monster that you summoned. The feat doesn't require adjacency, so with this feat you're also flanking in scenario 3 as long as either ally is a monster that you summoned; this, too, renders the other ally irrelevant for flanking purposes and also eliminates scenario 3's need for the need for stance island of blades (Tome of Battle 37). The Despana School feat's prerequisite is a base attack bonus of +5, the ability to cast a summon monster spell, and the feats Power Attack (Player's Handbook 98) and Weapon Focus (PH 102), with the latter applied to either the light mace, heave mace, or warmace (Dungeon #133 80 superseding Complete Warrior 154, 158) (25 gp; 10 lbs.). You were building around this, right?
The general feat that's also a fighter bonus feat Double Team (Dragon Compendium Volume 1 95—6 and not to be confused with the totally different feat of the same name from the earlier Dragon #357 39) allows you to remove either ally from the diagram and still leave you and the ally flanking the foe if both you and that ally have this feat. This feat also doesn't require adjacency, so in scenario 3 you're also flanking if both you and either ally have this feat; in scenario 3, again, the other ally is rendered irrelevant for flanking purposes and the island of blades stance remains unnecessary. The Double Team feat's prerequisite is an excessive base attack bonus of +6 and the feat Combat Reflexes (Player's Handbook 92).
If any route in this answer can be recommended, it's probably this feat. The 2nd-level Sor/Wiz spell heroics [trans] (Spell Compendium 113) can grant the feat to folks who meet its prerequisite so they don't have to actually take it. Heck, if the party has the heroics spell available, you don't have to take the Double Team feat. It's not like it's a prerequisite for anything.
The Rokugan ancestor (Crab) feat Great Teamwork (Oriental Adventures 63) allows you to remove either ally from the diagram and still leave you and the ally flanking the foe, but this benefit only applies while you're adjacent, so it does not also apply to scenario 3. Note that "[a]ny character can choose an ancestor feat, but only at 1st level" (Dragon #318 37), and the feat otherwise has no prerequisite. A similar benefit is realized from the feat Adaptable Flanker (Player's Handbook II 71, 74), but its prerequisite is ridiculous and its benefit inferior.
Now, with that out of the way, so far as I'm aware, only one official Dungeons & Dragons 3.5 game element facilitates flanking in the way that's actually desired in scenario 1: the teamwork benefit superior flank (Dungeon Master's Guide II 194).
Acquiring teamwork benefits
Because they're pretty obscure, here's the skinny on teamwork benefits: To acquire a teamwork benefit (Dungeon Master's Guide II 189—90) the team leader must have at least 4 Hit Dice, have at least an Intelligence score of 3, must be able to communicate with other members of the team, and meet the teamwork benefit's leader prerequisite, and each team member must have at least 4 Hit Dice, have at least an Intelligence score of 3, and meet the teamwork benefit's team member prerequisite. Then the team of 2—8 members (including the leader) trains together for 2 weeks, and one or more teamwork benefits are gained (up to 1 per 4 HD of the lowest-HD team member). Four times per year for 1 week each time, the team must again train together or else teamwork benefits are lost.
However, teamwork benefits just take time—no money need be spent to get teamwork benefits. Further, the design is fairly smart: the entire team need not be present for a teamwork benefit to be realized. If Jake's PC is the team leader or a team member and Jake's PC is dead, in prison, or off terrorizing peasants, then the rest of the team can go adventuring and still realize teamwork benefits (at least until it comes time to maintain the teamwork benefit).
Also, explicitly allowed to be team members are animal companions, familiars, and special mounts, but, of course, of course, they must still meet any benefit's prerequisite.
Superior Flank
For all this time and trouble, the teamwork benefit superior flank grants the following as its primary benefit:
Hence the teamwork ability superior flank only works with specific allies, but it does the thing that's desired in scenario 1 and your ally doesn't have to be a summoned monster or waste a feat to hang out with you, and you don't have to be from Rokugan. As an added incentive, every member of the team can do what you wanted to do, too. The team leader prerequisite for superior flank is sneak attack +4d6, and the team member prerequisite is base attack bonus +3.
Scenario 2: "When my allies are flanking a foe, I count as flanking that foe with a ranged weapon."
Nothing does this directly. However, as a curiosity, the whisperknife prestige class's level 9 extraordinary ability ranged flank says
Got that? Okay. Scoot 5 ft. to the right. There. Good. Now, if you're a level 9 whisperknife—and I'm not judging you—, then you're kind of in scenario 3 instead. Good luck.
Scenario 3: "If my allies or I possess an exceptional ability, I count as flanking that foe with a reach weapon."
Scenario 1 already describes the Despana School and Double Team feats. They do this. There aren't further exceptions that I'm aware of.
Scenario X: "If I've got the monster's nose, my allies count as flanking that foe" and other unusual options
These are additional options that I thought might be of interest despite not meeting the above criteria.
The general feat and also fighter bonus feat Dedicated Nuisance (Dragon #343 93) allows you to make a melee touch that, if successful, "[u]ntil just before your next turn, all creatures adjacent to the target other than yourself are considered to be flanking the target." This benefit is mind-affecting so you can't play I've Got Your Nose with vampires or whatever. Ask the DM if this melee touch attack must be made exclusively to activate the feat or if it can be any ol' melee touch attack. (I suspect it should probably be the former, but I think the latter's at least balanced.) Note that you must still conventionally flank a foe whose nose you stole if you want to flank that foe; this feat only helps your allies. The Dedicated Nuisance feat's ridiculous prerequisite is an Intelligence score of 13, a base attack bonus of +4, and the feat Combat Expertise (Player's Handbook 92).
The tactical feat Clarion Commander (Tome of Battle 34) allows you to use the tactical maneuver perpetual flank that says
In all but the most unusual encounters, successfully executing this tactical maneuver means that its benefit lasts until that encounter ends. The standard action that's required for the setup is an extremely high price, though, and a foe can foil your attempt in a variety of ways even before you miss with next turn's attacks, but if you can pull it off then the tactical maneuver perpetual flank may turn the tide of the battle for the right group—like a band of rogues, for example. The Clarion Commander feat's prerequisite is a base attack bonus of +6, two White Raven maneuvers, and the feat White Raven Defense (33–4) that itself requires a White Raven stance.
If both you and an ally have the technique (ronin) feat Serpent's Coil (Way of the Thief 47) and you and that ally flank the same foe, until the two of you aren't flanking that foe, that foe loses its Dexterity bonus to Armor Class. No, you're not misreading, and welcome to AEG's Oriental Adventures line. The foe loses its Dexterity bonus to AC against everyone and against every attack. You and your allies can even still get in there and flank that foe, too, if you want. If your other allies specifically need flanking, though, this is no help, but in the majority of cases—and, let's be honest, like you said, that's dealing sneak attack damage—, this is as good as flanking if not better.
Although the Serpent's Coil feat's prerequisite is only a base attack bonus of +4, that your allies also have to take it makes the actual price much higher. Even a DM who allows material from Way of the Thief may balk at allowing an animal companion or similar buddy to take the feat: the creature that takes the feat is supposed to be associated with the Serpents of Sanada organization described therein. Maybe they'll let your animal companion join? You can explain how he's a team player.
Note: This answer is for Dungeons & Dragons 3.5 exclusively and is not meant to be comprehensive. The writer has deliberately ignored Pathfinder material.