To say that a category "consists of the data" means that in order to specify a category, one has to tell their reader exactly the data that lies inside the definition. For instance, a monoid consists of the following data: A set $M$; a binary operation $\circ:M \times M \to M$ that is associative, i.e., $\circ(g,\circ(h,k)) = \circ(\circ(g,h),k)$; and a distinguished element $e \in M$ called the identity that has the property $\circ(g,e) = g = \circ(e,g)$ for all $g \in M$. To give a monoid $M$ is to define all of these things simultaneously. Thus to say what a category consists of is to define what makes up a category. In this sense to say that something belongs to a category, we mean that it is an element of the object "set," has a corresponding identity morphism in the morphism "set," and satisfies the necessary identities with respect to composition. Here we need some notion of "set," but whatever theory you wish to work in will be appropriate; for instance, you can use ZFC, you can use some set theory involving some appropriate notion of a Grothendieck universe in which every "set" smaller than some fixed strongly inaccessable cardinal $\kappa$ is "small" and every other "set" is large, or even a strange model of set theory. The beauty of the elementary axioms of category theory is that they hold so long as your set theory can handle the quantification and universal instantiation that is necessary to give the required data. Note that we can move past the notion of a primitive universe, say such as the category $\mathbf{Set}$, by then moving through an enrichment process, i.e., we add to our set theory all the things we need to define a universe in which our theory makes sense. If you are interested in this theory, you may want to read some enriched category theory and learn about $\mathscr{V}$-enriched universes. Note that when you enrich your set theory, even though you have, say, a new way of sovling the halting problem, you now have a $\mathscr{V}$-halting problem that you will have to enrich again to fix. This problem occurs infinitely, and needs some sort of transfinite induction to really talk about precisely.
When Awodey talks about the collection of objects and the collection of morphisms, he means a "Set" in whatever relevant set theory you care about for the moment. For instance, when we define the category $\mathbf{Set}$, the collection of objects is not, in the sense of ZFC, a set: It is a proper class! However, in this case we simply acknowledge the fact that sometimes we need to move up a universe of set theory in order to work. Again we run into the topic of enrichment, which makes this topic precise. Many people in math, myself included, just take for granted that we live in some Grothendieck universe and eventually have to move into dealing with "large" categories. These categories generally behave strangely, but are extremely important. For instance, in algebraic geometry, if $X = (X,\mathcal{O}_X)$ is a scheme, then the category $\mathbf{\mathcal{O}_X-Mod}$ is (generally) a large Abelian category, while for any ring $R$, the category of left $R$-modules with nondegenerate $R$ actions is a very nice Abelian category, al. In fact, the celebrated Mitchell-Freyd Embedding Theorem tells us that if $\mathfrak{A}$ is a small Abelian category, then there is a full and faithful embedding of $\mathfrak{A} \to \mathbf{R-Mod}$ for some ring of unit $R$ that preserves exactness. This gives us the result that essentially means that if you are arguing with a finite diagram in an Abelian category $\mathfrak{A}$, then you may as well use modules and module theory in order to make your life easier. For the most part you can treat the object "Set" and morphism "Set" of a category as if they were sets, or elements of some enriched and larger theory where such things make syntactic sense. Just be careful: objects in a category need not be sets, and you cannot argue set theoretically about some things. For instance, if we are dealing with a large Abelian category $\mathfrak{A}$ and we have an infinite diagram that we need to deal with, such as a cohomology long exact sequence, then we CANNOT, at least in general, use Mitchell-Freyd and we must argue with universal properties and such, save for special cases.
To address your third question, the difference between data and laws is that laws, in this sense, are syntactic or algebraic rules that MUST hold at all times, while data can vary. For an example, note that the categories $\mathbf{FinSet}$ of finite sets and $\mathbf{FinGrp}$ of finite groups have varying data: the morphisms in $\mathbf{FinSet}$ are very different from the morphisms in $\mathbf{FinGrp}$; however, the morphisms satisfy the same basic composition laws ($f:A \to B$ and $g:B \to C$ gives a composition $g \circ f:A \to C$), the same associativity laws ($f \circ (g \circ h) = (f \circ g) \circ h)$, and the same unit laws ($f \circ \operatorname{id} = f, \operatorname{id} \circ f = f$)!
