Are quantum states unit vectors in the space defined by their basis vectors? Also, if this is true, does this mean that the possible values of some quantum state are geometrically represented by unit spheres of $n$-dimensions and does thinking of these quantum states in this geometric way give us special ability to think about questions that other methods don't afford us?
 A: In addtion to the comment by ZeroTheHero, when referring to quantum states, to be more accurate we talk about rays in the Hilbert space, and not usually vectors (or unit vectors as you stated). Remember that a ray in Hilbert space is the collection of vectors that differ only by a phase factor.
We can however talk about "unit rays" as oppose to unit vectors, in which case a unit ray in a Hilbert space is the set  $$\{\lambda \psi : \lambda \in \mathbb C : \lambda\ne 0 \ \text{and}\ \ |\lambda|=1\}$$ where $\lambda$ is an arbitrary complex number (with unit magnitude).
An important point to note is that the magnitude, or length of a state in a Hilbert space is not physically distinct from a state that is longer or shorter (their relative "directions" are relevant, or value of their inner products).
A: The answer to the titular question is certaintly yes (I assume you're thinking about pure states). As to the follow-up, a complex $n$-dimensional unit vector indeed corresponds to the surface of a $2n$-dimensional sphere, but as per @ZeroTheHero I think this seldom helps us think about questions. One case in which it does help is in considering random unitaries acting on the state, which correspond to taking a random unit vector on the surface of such as sphere, so then we get to use the mathematical machinery of random matrices and SU$(n)$.
