Thus, for example, the proposition "x and y are numbers implies $(x+y)^2 = x^2 + 2xy + y^2$ " still holds equally if for x and y we substitute Socrates and Plato[2]: both hypothesis and consequent, in this case, will be false, but the implication will still be true. Thus in every proposition of pure mathematics, when fully stated, the variables have an absolutely unrestricted field: any conceivable entity may be substituted for any one of our variables without impairing the truth of our proposition.
By W&R's type theory, a number is a class of classes while Socrates and Plato are individuals. If we substitute Socrates and Plato for x and y, both the hypothesis and the consequent will be nonsense. E.g., $\alpha \not\in \alpha $ is neither true nor false: it is nonsense. And if $a$ is an individual, $\lambda$ is a class of classes, then $a\not\in\lambda$ is also nonsense. Thus, both "Socrates and Plato are numbers," and "$(Socrates + Plato)^2 = Socrates^2 + 2Socrates Plato + Plato^2$" are nonsense.
When $\phi \supset \psi$ is asserted, neither $\phi$ nor $\psi$ needs to be true, but both has to be significant. Thus, the variables in $\phi$ and $\psi$ must be restricted within their own types. The last sentence of §7 states that "the variables have an absolutely unrestricted field." I think W&R must have abandoned this view by the time they finished Principia. Russell's introduction to the 2nd ed of The Principles didn't point this out, please let me know if I am mistaken at this point.
You are right : Russell's view before the developmen of the Type thoery as a solution to the paradoxes was exactly that the logic was a sort of "science of everything", i.e. a set of laws so general to be applicable to absolutely "everything".
Thus, the "individual" variables of what we called today predicate logic must range on every object whatever : human beings, numbers, etc.
See e.g. :