There seem to be two parts to your question: one concerning whether all quantification is restricted, and another concerning whether free variables can carry contextual information that serves to restrict their admissible valuations. I'll address each in turn.
It seems like your question is regarding the possibility of completely unrestricted quantification. The mainstream view is that such unrestricted quantification over everything is possible. In practice, when doing model theory we often construct models with determinate (restricted) domains to show model existence and consistency of a theory by way of its satisfiability.
This model-theoretic convention, though, doesn't answer the question of whether all quantification must be restricted. The doubter of unrestricted quantification holds a view termed restrictivism. Typically, restrictivists derive motivation for their view from various set-theoretic and property-theoretic paradoxes. They claim that unrestricted quantification over, e.g., sets is impossible because the domain of quantification would be the set of all sets, which leads to paradox in standard set theories. Given the standard move of allowing proper classes to serve as a domain, or opting for a plural logic interpretation of the domain as the "plurality of all sets", this seems like a case of taking the semantics too seriously. That domains are sets is an artifact of model theory but not a necessary postulate.
More sophisticated restrictivists realize that with suitable large cardinal axioms, what was a proper class without such large cardinals becomes a set. They typically complain, then, that the notion of set is indefinitely extensible (an objection popularized by Michael Dummett, who used the notion to push for intuitionistic logic/mathematics). The idea is that anytime you think you have "all" of the sets, you could always admit more by accepting some larger cardinal and extending the height of the cumulative hierarchy. They argue that since there's no non-arbitrary stopping point, and this process can continue on indefinitely, we are never in a position to quantify over a determinate domain of all sets. Such a view, however, seems to tie our abilities to quantify over a domain to our ability to describe that domain uniquely (at least up to isomorphism) -- something the Platonist will surely deny.
The more common response to the paradoxes (though the restrictivist will complain such a response is objectionably ad hoc) is to say that they only show us that some predicates (like "set of all sets") do not have sets as extensions -- not that unrestricted quantification over sets is impossible. (This is more or less the idea behind Zermelo's approach in formulating the axiom of separation. Such predicates are rules out since you must always "separate" a set from a set -- ensuring the extensions of any predicates used in instances of the axioms are sets.)
Additionally, restrictivism is often claimed to suffer from a paradox of self-defeat akin to the flaw afflicting the verificationist theory of meaning. There seems to be no way to state the view without rendering it false. Consider:
(R) Every quantified statement is restricted.
(Typically they'd claim this holds of necessity, but I've eliminated any modal notions for simplicity and because the additional complications don't help the restrictivist.) Now there is a dilemma:
- The use of "every" in (R) is restricted, and so (R) does not capture the restrictivist's view -- it's compatible with instances of unrestricted quantification that fall outside the scope of the restricted "every".
- The use of "every" in (R) is unrestricted, and so (R) is false since it provides its own counterexample.
How to coherently state restrictivism is a subject of controversy, with many dismissing the view as incoherent. The primary source for this topic is Rayo and Uzquiano's (eds.) Absolute Generality.
Contextually Restricted Free Variables
The second question is largely separate from the first. There are a few uses of such contextually restricted free variables, off the top of my head these divide into (at least) two major categories: Dynamic Semantics and (some) presentations of Quantified Modal Logic. I'll start with dynamic semantics.
In dynamic semantics the central idea is that context serves to restrict the admissible variable assignments. It introduces quantification over variable assignment functions. Atomic formulae are of the form P(x) and are interpreted as the set of all assignments which assign to x an object that is P. Existential quantification acts to "reset" the set of admissible assignments to be those that differ from the original set of assignments by a renaming of the relevant variable/constant according to the (usually newly, otherwise the renaming is the trivial identity) introduced variable bound by the existential quantifier. This is the approach of Dynamic Predicate Logic, which takes these "actions" on assignments to provide the meaning of quantified statements (hence "dynamic").
A different formulation of dynamic semantics is given by Discourse Representation Theory, tracing back to work of Lauri Kartunnen from the 60's (1968?) but finding more complete development in the early 80's in the work of Irene Heim and the work of Hans Kamp (Kamp's work and its descendants are what is usually meant in modern references to "discourse representation theory"). The main innovation of DRT was to introduce the notion of "discourse referents" to treat phenomena like anaphoric reference. Anaphora occurs when an expression (typically a pronoun) depends for its reference on some previously introduced individual -- a referent derived from the preceding discourse. Consider:
A man walked into the bar. He asked for a drink.
The idea is that the phrase "a man" introduced some particular but underspecified individual into the discourse and that "he" refers to this individual. This is treated via "discourse representation structures" (DRSs).
The first sentence is given the following DRS:
[x: Man(x), Walked-into-bar(x)]
Here, the variable "x" denotes a discourse referent introduced by "a man". Note that this is a particularly individual and so not simply what would be conveyed by an existential statement.
The second sentence has the following DRS:
(With the understanding that the gendered pronoun supplies additional information marking x as male. I've put x in bold to indicated that it's unresolved and must be identified with a previously introduced discourse referent.)
These two DRSs are joined via the operation of "merge" to produce the DRS for the whole discourse (the two sentences):
[x, x: Man(x), Walked-into-bar(x), Ordered-a-drink(x), x = x]
The discourse referent "x" is often glossed as a free variable whose value is supplied by context (i.e., context restricts the admissible assignments).
Quantified Modal Logic
Finally, some presentation of quantified modal logic -- notably Fitting and Mendelsohn's First-Order Modal Logic -- use contextually restricted free variables, under the name of parameters, to deal with names in modal discourse. A parameter differs from an ordinary free variable in that it is never allowed to be bound by a quantifier in the course of a proof -- its value is always supplied via contextual restriction on admissible assignments.
This is similar to the use of "dummy constants" in natural deduction systems for predicate logic. When you move from an existential claiming that "Something is P" to the claim that P(a), you are required to use a "fresh constant" -- a constant not previously introduced in your proof (without having been already discharged). The value of a here isn't really constant at all, it denotes some arbitrary thing that is P and is indifferent as to which P you choose. P then serves to restrict the assignments to a and in some systems free variables with such restrictions are used in place of dummy constants.
So, to conclude, it is possible to assume that free variables "carry contextual information", even across a change in context, but the details of how this works will depend on the particular framework. Certainly it is a common phenomenon, even if not usually explicitly treated in introductory discussions of predicate logic.
As to unrestricted quantification, the majority view is that it is possible and coherent. The view does, however, have its detractors. It tends to be treated as almost a matter of dogma that unrestricted quantification is coherent and unproblematic, with the burden being placed on detractors to argue against it and formulate a coherent alternative.