Concerning the edges, consider the following:
Let $v\in \{0,1\}^d$ be a 0/1-vector with exactly $k$ zeroes. Then there exist $2^k$ 0/1-vectors that have vanishing inner product with $v$. The number of edges then is
$$\frac 12 \sum_{k=0}^d {d\choose k} 2^k = \frac {3^d}2.$$