I wasn't going to respond, however:
It is difficult metaphysically to draw the line between what is a property and what is an object. Some people argue yes. If existence is an abstraction and not something concrete, then to do so is a reification fallacy. However, in the SEP's article on abstract objects the argument is made that perhaps to consider a thing as both concrete and abstract at the same time we avoid the false dilemma which is another fallacy. The truth is you can manufacture either fallacy as a function of your metaphysical presuppositions, a hallmark of underdetermination Quine suggests we are constrained by when using evidence and language to describe our experience.
To complicate the matter, as Kristian notes, the problem of universals rears its head. Is a dog a concrete thing, or is it an abstraction over the breeds of dogs. Is a breed a concrete thing or an abstraction over instances of a breed? Is Fido a concrete thing, or Fido over some duration of time a separate set of concrete things? One of the ideas that Kristian has turned me onto is the notion of tropes (SEP): –
"Recently, however, both friends and foes of tropes have started to question whether tropes can be a bit of both. At any rate, this will depend on what being a property and being an object amounts to, an issue on which there is no clear consensus."
This is where metaphysical presupposition arises. What DOES it mean on being a property? What DOES it mean to be an object? Different people have different thoughts on that. Mereological nihilists admit no objects other than particles and configurations, for instance, and insist there are only configurations of atoms as chairs, and not chairs!
So, you asked:
Is there anything more fundamental than quantification?
As someone who cleaves to a constructionist epistemology, one possible solution would be to answer in the affirmitive. Yes, more fundamental than quantification is thought, because thought is that by which quantification occurs. This would be consistent with a reading of a nominalist (SEP) of abstract objects. Existential quantification is an abstract object which doesn't exist in the sense of apples and oranges. Rather it is an experience wherein we sense and use language to describe, and that the act proper is not fundamental. It's just one of a range of tools or experiences that allow us to organize our thoughts using language. This is why you'll see fiction used as a metaphor in describing certain positions. Just because we use words doesn't mean we are dealing with physical reality at all. Existential quantification is just vocabulary to describe being devoid of whatever it means to exist in a real sense.