The Quincies from Bleach set themselves up as being the opposites to the Shinigami in every way, and bear this out by being the only bow and arrow fighters in a series full to the brim with swords. Uryu Ishida (and his father Ryuken) also hits nearly every aspect of the Archer Archetype, being a tall, thin, aloof loner with a highly analytical fighting style. The Quincies normally favoring a ranged weapon despite being theoretically capable of creating any kind of weapon with reishi as demonstrated by the Vandenreich Quincies makes perfect sense when it's revealed that they have no resistance to Hollow attacks. The very essence of the Hollows is soul-consuming poison to the Quincies. Of course the Quincies would favor a means of destroying Hollows that doesn't require getting anywhere near them.
Rei/Sailor Mars of Sailor Moon gains a bow and arrow made of flames in the fourth season. She's a calm, mystical miko, rather than the typical temperamental fire warrior (in the manga at least, the anime makes her more Hot-Blooded), although she occasionally made bids for the leadership (losing out to The Fool, at that).
Rowen Of Strata (Tenkuu No Touma) is the archer of the Ronin Warriors/Yoroiden Samurai Troopers core team.
Gamaran has the Nakaizumi Ryuu (Hidden Spring School) which is composed by skilled archers. Their leader Arata went close to kill the titular character with his arrows. Another bow-wielding character is Ippi Shibano of the Muhou Ryu. While Arata represents the "good" traits related to archers (he's calm, really dedicated to his style, cool-headed and, unlike many archers, really muscular), Shibano is the "evil archer", a cowardly and cunning sniper who relies on surprise attacks and poisoned arrows.
Kikyou from InuYasha is a good example. She travels alone and is a little cold. As an archer, she is portrayed as a perfect shot and can deal devastating damage with her sacred arrow, an arrow surrounded with spiritual power that is basically a nuke. She is a little on the slow side, however, and is much more powerful long ranged than close ranged.
On the other hand, Kagome, Kikyou's reincarnation, travels with a ton of close friends and is almost always portrayed as warm and inviting. She is initially a horrible shot and doesn't have anywhere near as much training or power compared to Kikyou, but the gap closes quite a bit near the end.
Yes! Pretty Cure 5 GO!GO! has Cure Aqua with her new finishing move Sapphire Arrow, summoning a water bow and a water arrow. In case you didn't know, Cure Aqua is stated to be the Cure of Intelligence and she's usually being both The Lancer AND The Smart Guy in the group, keeping her calm most of the time.
Cure Aqua's Expy Cure Beauty from Smile Pretty Cure! has the ice variant of this, which is also her strongest attack, called Beauty Blizzard Arrow. In civil, she practices Aikido and hits the bullseye most of the time. And like her predecessor, she's calm and analytical, considered as the 'brain' of the group.
Seira from Il Sole Penetra Le Illusioni hit pretty much every point listed except for the build. She's the strongest of the group, as she's the only one that really trains.
Archers in The DCU come in a wide variety of flavors. Green Arrow (Oliver Queen) and Speedy (or Arsenal or Red Arrow or whatever they're calling him now) are both charming rogues, while the second Green Arrow is (usually) a cooler head. The crossbow-wielding Huntress is an Anti-Hero. Speedy II (Mia Dearden) is a more typical calm, level-headed example.
Hawkeye is an arrogant ace. He notably averts the "lower strength" trope, observing in at least one unarmed brawl how all the years of archery have paid off in terms of upper body strength and on another occasion disparaging a villain who's gotten his hands on his bow (which has a 250 pounds-force (1,100 newtons) draw weight) and can't even draw the string back far enough to use it.
Kate Bishop (the Hawkeye from Young Avengers) is a more typical example of a calm, level-headed archer.
Oxbow, of the First Line in Marvel: The Lost Generation, is a fairly huge subversion of the norm; he's 7 feet tall, super strong, easily angered, and not much of a braggart.
Strongbow (obviously) and Nightfall in ElfQuest, both of whom are fairly bold and not particularly weak.
In Marvel's G.I. Joe series, Storm Shadow is considered the greatest archer, even using it back when he was a GI in the Vietnam War. This is the one martial art where even Snake-Eyes could not equal or surpass him. Like most ninja in that series, he is enigmatic, aloof, self-assured, and rather irreverent at times. Zartan is the second-best archer, though he used various sonic detection devices to hit distant targets blindly. http://obsessionarchery.blogspot.com/2013/02/archery-terms.html
Films — Animation
Prince Ashitaka, the protagonist of Princess Mononoke is a skilled swordsman and hand-to-hand fighter but his REALLY badass feats are performed with his bow. He's a calculating fighter, but subverts the trope thanks to his messianic hero complex.
In Musa (a.k.a. The Warrior), Jin-lip, the veteran sergeant of the envoy, uses a bow to deadly effect. Jin-lip is the most competent member of the envoy and its de facto leader. He subverts an archer's typical behavior in one scene where he inspires his unit to attack by being the first one to charge toward the enemy, loosing arrows all the way.
Legolas in The Lord of the Rings has an incredible rate of fire with his arrows, and even uses the point-blank. This is even an exaggeration of his skill over his depiction in the books. Like all elves in the films he's elegant, graceful, and a bit snooty.
Katniss Everdeen in The Hunger Games is a masterful archer who relies on her bow to hunt, survive, and lead. Katniss has strong and independent survivalist instincts due to her difficult past and is good at thinking outside the box. She is not socially adept and has a hard time making friends due to the emotional strain on her life which has made her hard and cold. She is usually very logical.
Susan Pevensie is a graceful, elegant, ladylike kind of archer. Her bow was a gift from Father Christmas, and she became a famously skilled as Queen. One of the ways the Pevensies prove their identities to Trumpkin in Prince Caspian is by Susan beating him in a target-shooting contest. However, though quite effective in battle, Susan hates to fight.
In contrast, Lucy only trained in archery after growing into Queen Lucy the Valiant, but is a "tomboy" who fights alongside the men. For example, she joins the climactic battle in The Horse and His Boy, where Corin explains to Shasta that as a Queen she can do whatever she wants, whether or not she's female.
Catti-brie, of R.A. Salvatore's The Legend of Drizzt books, was for a long time the group's specialist archer. She actually wasn't as skilled an archer as Drizzt was, but she had one helluva magic bow. She conforms to the archer stereotypes by being the group's moral center of sorts.
Birgitte Silverbow from The Wheel of Time series. A literally legendary shot that is bound to the Wheel and the Horn of Valere until she ends up being "spun out" into real time by an angry Forsaken for saving Nynaeve. She has the cold analytical traits described here in spades, to such a high degree that others linked to her mind can sense her supreme focus in tense situations, and compare it to "a drawn bow aimed at the target."
Woodcrafters in the Codex Alera series tend to favor bows, since their abilities give them Improbable Aiming Skills. The most dangerous woodcrafters are those who also possess talent at earthcrafting, as this grants them Super Strength, allowing them to heft and use bows of such power that they're basically walking ballistae that can thread shots between links of chainmail. Amusingly, the two most prominent ones are a study in the opposite personality sides of this trope: Bernard, Supporting Leader and general great guy, and Fidelias, who is... well. Ratherdifferent.
Tortall Universe: In Trickster's Choice and Trickster's Queen, Sarai is portrayed as headstrong and careless, and uses a sword, while Dove is quiet and clever, and uses a bow and arrow.
Quantum Gravity: Zal fits this trope very well, being cool, calm, and calculating, as well as rather ruthless...when he's in a fight. When he isn't, expect him to be active, free, and quite a bit more easygoing. Aaaand possibly high.
In his child-rearing treatise ╔mile, ou de l'Úducation, 18th-century philosopher Jean-Jacques Rousseau notes that some professions are useful but dishonourable, requiring personality traits that are "odious and incompatible with humanity." He cites three examples: archers, spies and executioners.
J. R. R. Tolkien's Middle-earth stories feature far fewer stereotypical elvish archers than some copycat fantasy. The only character who fits the Archer Archetype is Beleg Strongbow (The Silmarillion, The Children of H˙rin). He's a famous Grey Elven archer and border-ranger captain, and quite capable of taking care of himself alone in the wilderness. His training in woodcraft is excellent, and he demonstrates his stealth several times, such as sneaking up on the outlaw band and tracking them through the wild. Later he sneaks into the orc camp and rescues T˙rin without alerting the orcs, after silently sniping the wolf guards one by one in the dark.
Mahabharata: Arjuna (nope, that does NOT mean archer) and Karna. (And those two are only the really big shots.)
Revolution: Charlie Matheson starts out as this with a crossbow. However, she has mostly stopped using it in favour of guns around episode 10.
Earthdawn naturally has a discipline for this, embodying the mentality and many of the tropes described at the top of the page.
Dungeons & Dragons has the Ranger, a subclass of fighter originally, but it became its own class in 2e. They eventually acquired a measure of magical ability, gaining the power to cast some divine spells. In 3e, there was also the Scout, perhaps a straighter version of this trope.
Elves are naturally known for this in the Warcraft Universe, particularly the more feral Night Elves. The Priestess of the Moon in particular from Warcraft III (and the Dark Ranger, an undead High Elf) put their bowstrings to very good use on fleshy opposing armies. Class is a bigger variable than race in World of Warcraft. Though bows are available to a number of classes, only Hunters possess the skills and talents necessary to make them into effective weapons. Good Hunters also embody the analytical traits described here — not only does their long-range position and lack of flashy effects (compared the nukers like mages and warlocks) give them a much better vantage point of a pitched battle, but they are masters of battlefield control with aggro management skills like Distracting Shot and Feign Death, the ability to set traps, and being able to act in two places at once by commanding their pet.
Lady Sylvanas Windrunner, Banshee Queen of the Forsaken, is regarded as one of the four best archers on Azeroth. She, her two High Elf sisters, and the Night Elf Shandris Feathermoon are considered to be equal with the bow and the best four archers on Azeroth.
The Naga Lady Vashj is also a superb archer, though more magically focused than most Archers.
Trine's resident thief, Zoya, is equipped with a bow that when properly upgraded, is Game Breaker in combat. She initially plays the personality part straight, but becomes more talkative as the game goes on, especially when the main party visits the forest.
Lamia Loveless in Super Robot Wars Advance is a calm, analytical, calculating android excelling in robot piloting, and her stats usually gears more on the ranged stat. Therefore, her main mecha, Angelg, also comes equipped with a bow, and the majority of its ultimate attacks are from its bow (Illusion Arrow & Phantom Phoenix). Though occasionally subverted that due to a bug developed in her latter stages, she could end up very hyperactive and loud unlike most of the archetypes here, but usually reverts back to her usual calm, analytical, calculating type.
Wood Elves, or Bosmer, are also quite known for their archery skills in The Elder Scrolls games, such that many guides to creating an archer in Skyrim will recommend them as the race of preference for those seeking to create one.
The Fire Emblem series features bow-users as a staple class, with some of them falling into this trope more than others:
Sacred Stones: Prince Innes is a noted strategist who operates an intelligence network and hits all of the Jerk tropes—"aloof" is the nicest you'll get out of him, unless you're Vanessa or Syrene. One optional dialogue has Eirika telling him off for not staying in the back like he's supposed to.
Dragon Age II has Varric and Sebastian, both calm and level-headed archers who subvert the trope slightly in that Varric is a smooth-talking urbanite and Sebastian possesses a stubborn vengeful streak.
There are a lot of ranged characters in Suikoden III but Roland and Jacques fit the trope to a tee. Both start with a rune that allows for a powerful ranged attack and both are deadly marksmen who keep themselves to themselves and say very little.
Worm features Shadow Stalker - a Nominal Hero who can phase in and and out to pass through solid objects. She uses a bow to hunt her targets and she prefers to work alone.
Evangelyne of Wakfu is an archer and the most level-headed member of the group. Averted with her hot-headed sister Cleophee, who fights with a wrist-mounted crossbow.