As Gerhard Paseman has commented, more general problems were studied by D.H. Lehmer in his paper The distribution of totatives. In particular, see section 5 of the paper. From his work it follows that $D(n)$ is a multiplicative function defined on prime powers as follows: $D(p^k)=0$ if $p\equiv 1\pmod 4$ for all $k\ge 1$; $D(p^k)=-2$ if $p\equiv 3\pmod 4$ and for all $k\ge 1$; and finally $D(2)=-1$, and $D(2^k)=0$ for all $k\ge 2$. The proofs are based on the simple sieve of Eratosthenes, and clearly this formula establishes that all $D(n)$ are zero or a power of $2$ in magnitude; the other assertions in the problem also follow easily.

As Gerhard Paseman has commented, more general problems were studied by D.H. Lehmer in his paper The distribution of totatives. In particular, see section 5 of the paper. From his work it follows that $D(n)$ is a multiplicative function defined on prime powers as follows: $D(p^k)=0$ if $p\equiv 1\pmod 4$ for all $k\ge 1$; $D(p^k)=(-1)^k\times 2$ if $p\equiv 3\pmod 4$ and for all $k\ge 1$; and finally $D(2)=-1$, and $D(2^k)=0$ for all $k\ge 2$. The proofs are based on the simple sieve of Eratosthenes, and clearly this formula establishes that all $D(n)$ are zero or a power of $2$ in magnitude; the other assertions in the problem also follow easily.