Use adamantine arrows to bypass appropriate damage reduction

As you've noted, the primary purpose of the special material adamantine seems to be damaging objects, yet adamantine arrows are typically a lousy choice for, for example, destroying a dungeon wall. But that's not all the material's good for. The universal monster ability damage reduction says

Some monsters are vulnerable to piercing, bludgeoning, or slashing damage. Others are vulnerable to certain materials, such as adamantine, alchemical silver, or cold-forged iron. Attacks from weapons that are not of the correct type or made of the correct material have their damage reduced, although a high enhancement bonus can overcome some forms of damage reduction.

An archer will want a batch of adamantine arrows if, for example, confronted with a gemstone gargoyle or a creature that's the subject of the spell stoneskin. While some archers bypass DR/X (adamantine) using a bow with an enhancement bonus of +4 or better, not all archers are fortunate enough to have such a weapon.

The barbed arrows are woefully underwritten. Further, the special weapon feature grapple just doesn't play well when applied to ammunition.

Attaching the silk rope to the barbed arrow

According to barbed arrows, "When a barbed arrow is attached to a length of silk rope and fired from a bow, the arrow’s range increment is reduced to 30 feet, but it gains the grapple special weapon quality."

How exactly an archer attaches a length of silk rope to a barbed arrow is a minor mystery. In Pathfinder's antecedent dnd-3.5e, an archer would employ the skill Use Rope, a skill Pathfinder thankfully deleted yet did not neatly replace. As binding a foe is based on the binder's Combat Maneuver Bonus as per the skill Escape Artist and a grappler can take a standard action to tie up a pinned foe, presumably an archer can take a standard action to attach a length of silk rope to a barbed arrow, but this is speculation.

Attacking with a barbed arrow that has a silk rope attached

After the silk rope is attached, an archer lets the barbed arrow fly. On anything less than a critical hit against his foe, the archer deals normal damage. On a critical hit against his foe, the archer deals his critical hit damage and can take a free action to make a grapple attempt… despite the typical barbed arrow (like most ammunition) "that hits its target [being] destroyed or rendered useless" and the stubby length of the attached silk rope.

The archer must take that free action to make a grapple combat maneuver after the archer's dealt that critical hit damage else lose the opportunity to do so, but from there, the archer obeys all the rules for the special weapon feature grapple and the rules for executing the combat maneuver grapple. To put this in context, here's the combat maneuver grapple rewritten so that an archer can use it when he makes a successful critical hit with the barbed arrow:

As a free action, the archer can attempt to grapple a foe, hindering his combat options. He does not provoke an attack of opportunity from the target of this grapple. Humanoid creatures without two free hands (which will be most PC archers) that attempt to grapple a foe take a –4 penalty on the combat maneuver roll. If successful, both you and the target gain the grappled condition. If you successfully grapple a creature that is not adjacent to you, move that creature to an adjacent open space (if no space is available, your grapple fails). Although both creatures have the grappled condition, you can, as the creature that initiated the grapple, release the grapple as a free action, removing the condition from both you and the target. If you do not release the grapple, you must continue to make a check each round, as a standard action, to maintain the hold. If your target does not break the grapple, you get a +5 circumstance bonus on grapple checks made against the same target in subsequent rounds. Once you are grappling an opponent, a successful check allows you to continue grappling the foe, and also allows you to perform one of the actions (as part of the standard action spent to maintain the grapple) listed later in this section.

(Emphasis mine.) In other words, if the archer succeeds on that grapple combat maneuver, the archer pulls the foe adjacent to the archer then can opt to either continue the grapple or take a free action to release the grapple.

Now, I deliberately don't use the word clearly very often, but clearly the vast majority of archers do not want foes adjacent to them! I can only assume the (ahem) point of barbed arrows is not to grapple a foe, drag the foe adjacent to the archer, then have the archer proceed to rassel the crap out of that foe—most archers aren't just aren't specced for that, and giants that do this to PCs straight-up murder those poor PCs.

However, that's what happens. I mean, the weapon quality grapple says, "While you grapple the target with a grappling weapon, you can only move or damage the creature on your turn," and that makes sense, but then it continues, saying, "You are still considered grappled, though you do not have to be adjacent to the creature to continue the grapple" (emphasis mine), but that means a creature that's using a weapon with the grapple quality must still be adjacent to a foe to start the grapple! Then the description finishes, saying, "If you [in this case, the archer] move far enough away to be out of the weapon’s reach, you end the grapple with that action."

In other words, the special weapon feature grapple functions fine on reach weapons and other melee weapons that hook a foe, drag the foe over, and then allow the attacker to move away from the foe within the weapon's reach, but with a ranged weapon? Yuck. I mean, what's the reach on an archer's arrow? The archer's natural reach. What's the reach—not range increment, mind you, but reach— on an archer's bow? 0 ft. What's the reach on the archer's barbed arrow that's destroyed because it successfully struck a foe? That'd be a flat nothin'.

Questions and Answers

In short, not a whole lot of this makes sense. Keep this assessment in mind while I take each question in turn.

• Should a weapon that possesses the special weapon feature grapple and that possesses a magical bonus apply that bonus to the wielder's grapple checks to start a grapple? This GM would say yes, but this GM would have an archer's arrow be the weapon not the archer's bow.
• After a successful grapple, should a weapon that possesses the special weapon feature grapple and that possesses a magical bonus apply its bonus on grapple checks made to maintain the grapple? This GM would say yes. The wielder has employed the weapon to start the grapple and must keep the grappled creature within his weapon's reach. The weapon's helping with the grapple, so any magical bonuses should apply.
• Does the wielder of a weapon that possesses the special weapon feature grapple and that possesses a magic weapon special ability apply the effects of that magic weapon special ability when the wielder opts to deal damage during a grapple that the wielder initiated with that weapon? This GM would say yes, but only if that weapon is a one-handed or light weapon. For example, a +1 flaming barbed arrow could be used as an improvised light weapon, so that'd be legit.
• After making that successful critical hit and after making a successful grapple combat maneuver, if an archer has attacks remaining, can he make those attacks? No. A grappled creature can take no action that requires two hands to perform, and firing a bow takes two hands.
• Because both archer and foe have the grappled condition, is it accurate to say that the normal grapple effects can be used? No. A creature that's started a grapple by wielding a weapon with the special weapon feature grapple can only exercise the grapple options move or damage; other options are unavilable.
• A barbed arrow has a range increment of 30 ft. What about the attached rope? A barbed arrow—with or without a rope attached—has a range increment of 30 ft. (hence a maximum range of 300 ft. or 10 range increments). The attached silk rope—sold separately—only extends to the length of the rope, not—as some special feature of the barbed arrows—beyond the silk rope's purchased length. This GM assumes that the idea is that an archer can't attempt a grapple against a foe on whom he's scored a critical hit if the silk rope isn't long enough to span the space between them, but the rules don't mention this. Rules as written, even a 1-ft. length of rope is sufficient to—I dunno—mime-pull a foe adjacent to the archer after a critical hit and a successful grapple check, another silly result of an underwritten piece of gear.
• Could a foe cut the rope, break the arrow, or employ some other asymmetrical method to escape this grapple that doesn't involve an Escape Artist skill check or an opposed grapple check? Not according to the rules.

This GM strongly urges a house rule saying that an archer that makes a successful critical hit with a barbed arrow that has a silk rope attached instead renders his foe entangled much like a net, the archer controlling the trailing rope if the silk rope attached to the arrow is sufficient to span the distance between archer and foe and if the archer is willing to commit a free hand to the task.