To comment on the second part of your question:
When we talk about formal languages, we do this in another formal language, the so-called "meta-language". If you want to talk about the theory of the natural numbers (that is, the formulas satisfied by the structure of the natual numbers), you do this in a meta-language, usually ZFC, in which you can define the set of natural numbers, the set of formulas, and what it means for a formula to be true.
In classical logic, you just translate the universal quantifier of the language to the universal quantifier of the meta-language.
$\forall x \varphi(x)$ is true iff
$\varphi(t)$ is true for all
$t$. All we've done is defined the universal quantifier in the language using the universal quantifier in the meta-language.
So if you want to define a domain of discourse as
$\varphi$ can be any formula in the meta-language. So if you use higher-order logic as metalanguage,
$\varphi$ can be any higher-order formula.
As for syntactical rules: You do not need a domain of discourse in order to syntactically derive valid formulas, because all you do is manipulating strings. The introduction rule for universal quantification just says that if you can prove
$\varphi(x)$ (which means that the proof cannot depend on the value of
$x$), then you can prove
$\forall x \varphi(x)$. This is just adding two symbols at the beginning of a formula.