Oh, well, I'll try my best to help.

(0) Hypercube is not really super-nice, it is merely Lipschitz! This is of course a joke, because periodic boundary conditions indeed make it perfectly regular. But then...

(1) Even for the usual Laplace operator the Green function (in the strict sense: as the kernel of the inverse operator $(-\Delta_{\mathbb{T}})^{-1}$) is not defined, because $\Delta_{\mathbb{T}}$ has eigenvalue $0$ (corresponding to a constant eigenfunction $e_1$; I use ${\mathbb{T}}$ for the hypercube here and below). The same is clearly true for the fractional power $(-\Delta_{\mathbb{T}})^{\alpha/2}$, if it is defined via spectral theory (in this case: Fourier series), as in the second display in the statement of the question.

(2) Noteworthy, it does not matter whether we take the fractional power of $-\Delta$ first and then periodize (is this the right word?), or vice versa. The easiest way to see this is to note that Bochner's subordination formula for the heat kernel of a fractional power commutes with periodization, so to say.

(3) The inverse operator $G^{(\alpha)}_{\mathbb{T}} = (-\Delta_{\mathbb{T}})^{-\alpha/2}$ is well-defined on the orthogonal complement of constants. Of course it can be expressed in terms of spectral theory: $$G^{(\alpha)}_{\mathbb{T}} v = \sum_{k = 2}^\infty \lambda_k^{-\alpha/2} \langle v, e_k \rangle e_k ,$$ but there is a more useful expression: the kernel of $G$ is a "compensated" integral of the heat kernel: $$G^{(\alpha)}_{\mathbb{T}}(x, y) = \int_0^\infty (p^{(\alpha)}_{\mathbb{T}}(t, x, y) - 1) dt .$$ Here $p^{(\alpha)}_{\mathbb{T}}(t, x, y)$ is the heat kernel of $(-\Delta_{\mathbb{T}})^{\alpha/2}$. Indeed, in order to prove the above formula, one simply expands the heat kernel in terms of the eigenfunctions and uses Fubini.

(4) Since we have already observed that $p^{(\alpha)}_{\mathbb{T}}(t, x, y)$ is the periodization of the heat kernel $p^{(\alpha)}(t, x, y)$ in full space, it is not hard to see that the order of the singularity of $G^{(\alpha)}(x, y)$ at the diagonal is preserved. Namely,
$$G^{(\alpha)}_{\mathbb{T}}(x, y) - G^{(\alpha)}(x - y) = \int_0^\infty \left(\sum_{k \in \mathbb{Z}^d \setminus \{0\}} p^{(\alpha)}(t, x, y + k) - 1\right) dt$$
is uniformly bounded due to well-known bounds on the heat kernel.

(5) In fact the order of singularity of the Green function of $(-\Delta)^{\alpha/2}$ is the same in essentially every reasonable space. One only needs Gaussian (or even sub-Gaussian!) bounds for the heat kernel of $(-\Delta)^{\alpha/2}$ for small times, and the subordination formula. I suppose this can be found in Andrzej Stós's article *Symmetric $\alpha$-stable processes on $d$-sets*, which unfortunately does not seem to be easily available online.

Finally, let me stress that I do not really know the literature on fractional Laplace operators in spaces other than $\mathbb{R}^n$.