Lower bounds for nonuniform circuits and oracles separating complexity classes
Yes, yes, and yes. 

The basic idea is to consider the characteristic function of a language $L$ 
(the oracle you're constructing) at length $n$ as a string of length $2^n$ 
that will be an input to a ("big") circuit. 
An OR gate in that circuit will correspond to 
a polynomially-bounded existential quantifier, and 
an AND gate to a polynomially-bounded universal quantifier. 

Consider Furst-Saxe-Sipser as an example of 
how this can be used to build oracles:
If the language $L$ is constructed so that 
the characteristic function of $L \cap \Sigma^n$ is an input 
whose parity differs from an $\mathsf{AC}^0$ function 
(which exists, by Furst-Saxe-Sipser), 
then you've just successfully diagonalized against 
the language in $\mathsf{PH}$ corresponding to that $\mathsf{AC}^0$ function.
Doing this over all $\mathsf{AC}^0$ diagonalizes against all of $\mathsf{PH}$, and thus builds an oracle $L$ such that $\mathsf{PH}^L \neq \mathsf{PSPACE}^L$
(since the latter includes the language corresponding to 
the "big" parity circuit).

For more discussion and examples of this general theme see the following survey :


Ker-I Ko, "Constructing Oracles by Lower Bound Techniques for Circuits", 1989.