Is the zero vector in $\mathbb{R}^n$ by itself a subspace of $\mathbb{R}^n$? W is a subspace of $\mathbb{R}^n$ iff


*

*The zero vector ∈ W.

*X + Y ∈ W for any X, Y ∈ W.

*aX ∈ W for any X ∈ W and a ∈ R.


So, given W = { X : X = [x1...], x1 = 0, x2 = 0, ... xn = 0 } ∈ Rn


*

*The zero vector ∈ W because each X in W is the zero vector by definition,

*X + Y ∈ W because [0...] + [0...] = [0...]

*aX ∈ W because a[0...] = [0...]


So if I understand this correctly, the zero vector is itself a subspace of $\mathbb{R}^n$. Is this correct? 
In addition, can this be extended to say that for any W = { X : X = [$x_1$...] } ∈ $\mathbb{R}^n$, assuming W is a subspace of $\mathbb{R}^n$, $x_i$ (an arbitrary component of W) can only be a constant if it is 0? (I.e., $x_i$ can't be 1 or 2, but can be 0)

Indeed, $\{0\}$ is a subspace. In fact, for any vector space $V$ and any $x\in V$, we have that $\{x\}$ is a subspace of $V$ if and only if $x$ is the zero vector of $V$, since any subspace of $V$ must contain the zero vector of $V$, and the subset of $V$ containing only the zero vector of $V$ is necessarily closed under addition and scalar multiplication (as you've shown).
