Attaboy or thattaboy or that a boy is an informal expression of praise, approval, or encouragement of American origin, first attested 1909. As AHD has it:
interj. Informal Used to show encouragement or approval to a boy or man:
Attaboy! That's the way to hit a home run!
[Alteration of That's the boy!]
The OED also gives that origin, which seems plausible, though that's my boy as an exclamation is easier to find and remains in popular usage.
Boy is used in the affectionate or playful sense, and does not necessarily refer to a youth, or even necessarily a male. Still, attagirl is a natural alternative for females, attested from the 1920s.
As an actual interjection I rarely hear it except with children or pets, or in fairly informal settings. As a noun referring to a short statement of approbation, it is unexceptional. It is like kudos or cheers in my idiolect.
As to Shooter, given your teacher's readiness to use familiar language with students, I would say it is simply a familiarizing nickname intended to demonstrate affection or informality. It may be an obscure reference, or he may simply have liked the sound of it, but it is not attached to any particular region or meaning.
All manner of words are used in this way by all kinds of people. Off the top of my head, you'll find television dialogue where someone is addressed as champ, tiger, kiddo, chief, sport, sailor, ace, amigo, or cowboy — with absolutely zero relationship to their championship, tigrinity, maturity, authority, athleticism, nauticality, aerial lethality, friendship, or … vaquerage? Anyway, in real life I also know people who address everyone as senator or gunner or homes or bad boy.
There are no real rules dictated by the language, only by social convention. For instance, some women may not appreciate being called big fella or papa bear, and cupcake or dreamboat may be interpreted as diminutive, or insulting, or flirtatious except among very close peers. As tchrist cautions in the other question, these words are therefore fraught and the risk of misunderstanding or insult is extremely high— as we seem to have seen here.