Q:

Semantics of Properties - Are categories of extensions members or subsets?

For example:
"Cars have wheels."
If we take "have wheels" as a property of a set A, would cars as a category be an element of set A, or only a subset of A?

A:

Whether or not a subset is considered an element of the superset is a metaphysical presupposition in your set theory. In naive set theory, the relationship between sets as elements is not clear, because often the context treats elements and sets as objects and containers through conceptual metaphor. As such, certain implications arise from having containers in containers.
With the advent of Russell-Zermelo Paradox (RZP), the question of whether sets in sets leads to contradiction (it does) comes to the forefront specifically regarding sets being members of themselves. This of course breaks with our intuition as containers cannot contain themselves. To wit:

Some sets, such as the set of all teacups, are not members of themselves. Other sets, such as the set of all non-teacups, are members of themselves. Call the set of all sets that are not members of themselves “R.” If R is a member of itself, then by definition it must not be a member of itself. Similarly, if R is not a member of itself, then by definition it must be a member of itself.

Should the set of car (4-wheeled), which has the same criteria as the superset, be included in the set as an element? That depends on whether or not you want to allow or disallow the operation. Type theory (which is essentially a restriction on set theory regarding operations and relations) and category theory (which is essentially an abstraction of set theory) have differing approaches to having collections as elements. In response to RZP, besides type theory, which was invented to avoid the paradox, so too came axiomatic set theories, e.g. ZFC, which still allows sets to contain sets, albeit with restrictions to avoid the paradox.
To answer your question, can given your definitions the set "cars" belong to the set "4-wheeled vehicles". That depends on how you define "4-wheeled vehicles". Whether by intensional or extensional definition, you can include or exclude it as you see fit; to a large extent it depends on the purpose of your use of set theory. Sets as elements of sets is the norm with sets such as the power set and heredity sets serving examples of sets within sets.
EDIT
It should be noted that the question you are asking is one that occurs within the philosophy of mathematics. What is the nature of a set? Of elements? Of membership? Although not explicitly stated, set theory is not a monolithic theory, but rather a collection of theories based upon different ontologies. The comparison of those logics and theories is known as metalogic and metamathematics.

