Loooooooool! Being an archer, I can't help but laugh! Dear sweet unicorn, you can't grip a bowstring with your teeth in that manner, you'd have more succes turning your head sideways, or the bow sideways, and after that I still wouldn't suggest shooting a bow in that manner, unless you enjoy frequent visits with the dentist. You've got magic, just grip the handle and the string, no need to grip the limbs of the bow as well.
I feel that unicorns just starting to use the bow and arrow, would first begin their training by using their teeth to pull the string back. As they become more familiar with holding the bow steady with their magic, they can start training themselves to pull the string back with their magic, thereby fully enveloping the bow and the string. The advantages to this are numerous, such as not having any part of the bow blocking the unicorn's vision and also not having to be in close proximity to the bow.
Ultimately, a master archer of the unicorn race can use several bows simultaneously, be pin-point accurate (that's a given), and be able to shoot in any direction with a 360 degree cone of vision.
This summary only covers the bow aspect of the archer. As for ammunition, unicorns can summon different types of arrows (holy, dark, frozen, fire, etc.). They can also use their magic to make arrows explode, to make arrows go faster with the element of wind to pierce certain targets, or even make arrows curve and hit targets behind cover. The possibilities are certainly endless.
Bows don't actually work that way. There isn't any energy "stored" in the bow beforehand, like it's got some kind of internal reactor. Remember conservation of energy: all the force behind the arrow comes from the archer when the arrow is drawn. The archer applies force to the bow by pulling back on the string and pushing against the handle. This is two forces being applied relatively slowly, causing the limbs to bend and store the energy from the archer's arms. Then once the string is released, the limbs bend back to a resting state and pull the string and the arrow along with them. Furthermore, the reflexiveness of the wood causes the arrow to accelerate very quickly - faster than the archer can swing his arm.
Also remember: Force = Mass * Acceleration. The mass of the arrow is constant; it's a matter of how fast that mass is accelerated from a resting state. What makes a bow superior to throwing is the bow's ability to hold and *rapidly release* energy.
This situation has a unicorn magically pushing on the bow and pulling on the string with his teeth. Obviously, a unicorn has different capabilities than a human. So, then, the question is whether or not his magic can *accelerate* the arrow faster than the bow, since the mass will be the same either way. This cannot realistically be answered, however, so it is debatable.
Also another thing I'd like to point out: he's probably holding the bow incorrectly. If he's "gripping" the limbs of the bow with his magic - as opposed to just the handle - that force would likely interfere with the reflexive action of the bow, causing his shots to be weaker and shorter.
Wrong, there is in fact energy stored in the bow beforehand, it's called tension. If it did not have tension in the limbs of the bow, the string would be slack. It's because there is tension in the bow before pulling the string that there is so much energy transferred into the arrow when fired. If there were no tension, most of the energy created by the action of pulling the string back and applying more tension to the limbs would be drained into the string as the arrow is released instead of being dumped into the arrow. Any bow that is "stringed" is in fact storing energy via the tension in it's limbs (any bow worth anything, that is). The rest of what you say is true, and in fact I do point out the bit about holding the bow's limbs in a comment further above. And trust me, it is far easier to pull on a bow than it is to expend the amount of energy it would require to hurl an arrow with the same speed and force by hand.
When you pull on a bow string, you're not expending the same energy the bow puts out when it fires. You're merely activating the energy already stored in the bow through the tension in the limbs. This does still require a fair bit of energy on the part of the user, yes, but the energy yielded by the design of the bow is far greater than that expended by the user.