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Can you roll and confirm a critical hit on a rapid shot? Assuming you have all the required feats for those attacks? What about the extra arrow from a Hasted attack?
When you crit on the first attack with Manyshot, you don't crit on the manyshot arrow, correct? So you'd just add the crit damage with the regular non-crit damage for the second/manyshot arrow?
Thank you!
Edit: Also, I am mounted (ranger) with Snap Shot so I threaten AoO 5 ft with my bow, and my mount has Charge Through, Greater Overrun and related feats. Rules state:
"You can use ranged weapons while your mount is running (quadruple speed) at a –8 penalty. In either case, you make the attack roll when your mount has completed half its movement. You can make a full attack with a ranged weapon while your mount is moving. Likewise, you can take move actions normally."
So essentially, my mount can Charge, halfway through his movement to the enemy I release a full round attack, and by the time we arrive to the enemy my turn is essentially over - mount proceeds to attempt overrun, which if he does successfully procs an AoO UPON falling prone (not on getting up yet!) so my ranger can then take that AoO and shoot one arrow, yes? Then another AoO if he stands up on his turn?

Phoebus Alexandros |

Critical Hits: When you make an attack roll and get a natural 20 (the d20 shows 20), you hit regardless of your target's Armor Class, and you have scored a “threat,” meaning the hit might be a critical hit (or “crit”). To find out if it's a critical hit, you immediately make an attempt to “confirm” the critical hit—another attack roll with all the same modifiers as the attack roll you just made. If the confirmation roll also results in a hit against the target's AC, your original hit is a critical hit. (The critical roll just needs to hit to give you a crit, it doesn't need to come up 20 again.) If the confirmation roll is a miss, then your hit is just a regular hit.
So that's the general rule.
When making a full-attack action with a ranged weapon, you can fire one additional time this round at your highest bonus. All of your attack rolls take a –2 penalty when using Rapid Shot.
Rapid Shot simply imposes a penalty on an (extra) attack that you make. It doesn't introduce anything that changes the general rule on critical hits, which covers the attack action, the iterative attacks you get on a full attack action, attacks of opportunity, and extra attacks from conditions like being hasted. A 20th level fighter using a bow might shoot six arrows at an opponent (full BAB, rapid shot, hasted). If all six attack rolls are in his bow's critical range, hit the enemy's AC, and are confirmed, then he'll inflict six critical hits.
When making a full-attack action with a bow, your first attack fires two arrows. If the attack hits, both arrows hit. Apply precision-based damage (such as sneak attack) and critical hit damage only once for this attack. Damage bonuses from using a composite bow with a high Strength bonus apply to each arrow, as do other damage bonuses, such as a ranger's favored enemy bonus. Damage reduction and resistances apply separately to each arrow.
Manyshot qualifies that both arrows hit, and both arrows inflict their full normal damage (to include bonuses), but only one of them applies precision damage and critical hits. It is a specific instance where the general rules on attacks and critical hits are both modified.
EDIT:
You're confusing two different actions. You're quoting what your character can do while his mount is making a run action, and trying to apply a feat (Charge Through) that requires a charge action. These are two different things.
To be more specific, your mount isn't charging. The charge action (which allows you to move up to twice your speed) is distinct from a run action (which allows you to move up to four times your speed) or even a double move action (which allows you to move twice your speed, but not make a melee attack at the end). Greater overrun does allow your mount to make an overrun combat maneuver as part of its move (not just a charge), it just wouldn't be a free action.
Yes, halfway through that movement, you'd take a full attack action. Your mount would then complete its movement, and if your opponent is knocked prone by your mount's overrun combat maneuver, he would provoke an attack of opportunity. That attack of opportunity would be taken by your Ranger's mount, though, and not your Ranger... because Greater Overrun doesn't state the target provokes AOOs from allies as well. Your Ranger could take an attack of opportunity if your mount was still within five feet of your opponent when he got up. But that almost certainly means your mount would have to end its movement within five feet of the prone opponent.