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 :