We know that, given a set $X$, there exists at least one choice function $f:X\rightarrow\cup X$ thanks to the Axiom of Choice (AC).
Can we conclude that all choice functions for a generic set $X$ exist?
Do we have any information regarding the cardinality of the set of choice functions?
To an extent, this question doesn't make any sense.
The existence of an object is measured against a fixed universe. Namely, either there is or there is a choice function to begin with. And a set is either finite or empty or infinite or uncountable, or anything, in a given context and a fixed universe.
The set of all choice functions always exists. It is a consequence of how we define a choice function, as well as the other axioms of set theory. The axiom of choice merely implies that this set is not empty.
What is true, however, is that if we have a single choice function, provided that the family is not a family of singletons, there will be other choice functions as well. Indeed, we can combine all sort of choice functions together, and we can do all kind of "crazy things" to show that if there are two choice functions we can define a third (provided the family is not just one pair and a bunch of singletons), etc. But, this has nothing to do with the axiom of choice and everything to do with the other axioms of set theory.
For example, suppose that $\{A_n\mid n\in\Bbb N\}$ is a family of non-empty sets such that no infinitely many of them admit a choice function. And now consider $B_n=A_n\cup\{n\}$. Now $\{B_n\mid n\in\Bbb N\}$ does admit a choice function, namely $n\mapsto n$. But we can also alter this choice function on finitely many coordinates to pick members of the $A_n$s instead of the integer. We have a more complicated situation if an infinite subfamily does admit a choice function, too. But this is not because of the axiom of choice, this is merely because there exists a choice function to begin with. In the general the axiom of choice merely allows us to make that initial step, wherein the journey towards this chaos begins (and in a very deep sense, a journey from a much bigger and more complicated chaos also takes place).