This is not really an answer, but it's too long for a comment. If $n$ is a prime $p$, then you are asking for the number of points in $\mathbb{Z}_p^d$ not lying on any of the $2^d-1$ hyperplanes $x_{i_1}+\cdots + x_{i_k}=0$. By the general theory of hyperplane arrangements, for sufficiently large $p$ this number will be a polynomial in $p$ which is the characteristic polynomial of the corresponding real arrangement. The real arrangement is discussed in http://mathoverflow.net/62764. The problem of finding the characteristic polynomial (or even its value at $-1$, which is up to sign the number of regions) is considered to be intractable. Thus an exact formula for $\alpha_n^d$ (even when $n$ is prime) is highly unlikely. It still should be possible to obtain some reasonable estimates.

This is not really an answer, but it's too long for a comment. If $n$ is a prime $p$, then you are asking for the number of points in $\mathbb{Z}_p^d$ not lying on any of the $2^d-1$ hyperplanes $x_{i_1}+\cdots + x_{i_k}=0$. By the general theory of hyperplane arrangements, for sufficiently large $p$ this number will be a polynomial in $p$ which is the characteristic polynomial of the corresponding real arrangement. The real arrangement is discussed in https://mathoverflow.net/62764. The problem of finding the characteristic polynomial (or even its value at $-1$, which is up to sign the number of regions) is considered to be intractable. Thus an exact formula for $\alpha_n^d$ (even when $n$ is prime) is highly unlikely. It still should be possible to obtain some reasonable estimates.