These results belong to what is called duality in finite abelian groups, a theory that has been generalized by Pontryagin and others in the 30's to locally compact abelian groups.
Another keyword here is "Discrete Fourier Transform", although it is mainly used for cyclic groups (of order $2^N$ for FFT) or $GF(2)$ vector spaces (analysis of boolean functions).
It is also a chapter in the representation theory of finite groups, founded by Frobenius in the end of 19th century, namely the case of finite abelian groups, where the irreducible characters are all homomorphisms to $S^1$. This case was already known to Gauss and Dirichlet.
By the way, the orthogonal you define is more naturally seen as a subgroup of the dual group $\hat{G}=Hom(G,S^1)$, which in (multiplicative) duality with $G$ via the evaluation $G\times \hat{G}\to S^1$. It is isomorphic to $G$, but not naturally so.
I would bet that almost any book on either Pontryagin duality or representation theory of finite groups has the results you want to cite, but I have only been able to find this link to an online encyclopedy, which easily implies your claims, since 3 and 4 follow quickly from 1 and 2.
Hope this helps.