ZFC is meant to capture a certain notion, the cumulative hierarchy of sets. For this justification to make sense you need to think of "well ordering" or "ordinal" as a pre-existing mathematical concept, not one based on ZFC. This justification is explained at more length in Shoenfield's article in the Handbook of Mathematical Logic from 1977, and elsewhere. It dates back to the early 20th century.
Assuming that we have a collection of "ordinals" $O$, which is downward closed and has minimal element $0$, we can define a collection $V^O$ as follows.
Finally, $V^O$ itself is $\bigcup_{\alpha \in O} V^O(\alpha)$. The informal way to put this is: think of the elements of $O$ as "stages". Then a set will be put into $V^O$ at stage $\alpha$ if all of its elements have already appeared at stages earlier than $\alpha$.
We can ask: which axioms of set theory does $V^O$ satisfy?
$V^O$ will contain the empty set as long as $O$ has at least two elements, because the empty set appears in $V^O(1)$.
$V^O$ will satisfy the separation axiom. To see this, assume $z \in V^O$ and that we want to prove that $y = \{x \in z : \phi(x)\}$ is in $V^O$. Well, $z$ was formed at some stage, so all the elements of $z$ were formed at earlier stages. But this means that all the elements of $y$ were also formed at stages earlier than the one where $z$ was formed, so $y$ will be formed no later than $z$.
Every subset of a set $z$ is formed at the same time that $z$ is formed. Therefore, the powerset of $z$ will be formed at the stage after $z$ is formed, assuming there is a next stage. So if $O$ has no maximal element then $V^O$ satisfies the axiom of power set.
These examples suggest that the axioms that are satisfied by $V^O$ will depend on how "long" $O$ is. Indeed, it turns out that if we assume sufficient properties about $O$ then we can argue in a similar way that $V^O$ satisfies all the ZFC axioms. In particular, if we let $O$ contain all the ordinals, then $V^O$ will satisfy ZFC, and we usually just write $V$ instead of $V^O$ in this case. This $V$ based on all the ordinals will be a proper class, not a set.
This argument cannot be captured in ZFC itself, although various properties of the cumulative hierarchy can be captured in ZFC. But the argument does give some motivation for why ZFC should be consistent, by giving a conception of sets (as elements of $V$) which seems intuitively reasonable. Indeed, it appears that all we need to have in order to form $V$ is a well-determined collection of ordinals, the ability to take powersets, the ability to take unions, and the ability to iterate these operations along the ordinals.
So where could ZFC be inconsistent, even if this argument is correct? One place is the separation axiom. In the argument above we assumed that $\{x \in z : \phi(x)\}$ actually does define a subset of $z$ whenever $\phi$ is a formula of set theory. If somehow there were formulas $\phi$ which do not determine subsets of $z$, our argument for why the separation axiom holds in $V$ would not go through. There is a certain sense in which the argument above is proving the consistency of second-order ZFC rather than first-order ZFC, just as the informal proof of consistency of Peano arithmetic that says "$\mathbb{N}$ is a model" is really a consistency proof for second-order PA rather than first-order PA.
The simplest thing may be to just say what you mean. That is, if your theorem is that some statement is false, then just make that the theorem you're claiming. The only reason to display the false statement as you want is if there is going to be a protracted development before it is refuted or perhaps before you even begin to refute it. If you are going to immediately refute the statement and the proof is moderately short (say a page or less), then I would do this, i.e. state the actual theorem which is that the negation of the false statement holds.
You definitely should not label the false claim a theorem and then provide a counter-example instead of a proof.
"Proposition" would be fine as technically that doesn't imply any assertion of provability, though often it is taken to mean a true statement.
What I would recommend other than my first paragraph is "Claim", though you would still need to make it clear either immediately before or after that you are going to show that this claim is false, as "claim" doesn't have a connotation that you are going to refute it.1 This would be best if the fact that the claim is false is surprising. At that point, the structure of your text would be something like "A natural statement is . Surprisingly, we will show that it is false."
I've also seen "Non-Theorem" rarely which definitely would make it clear that you are asserting that it doesn't hold. When I've seen this, it's usually in more expository work that is pointing out statements that prior experience might naturally suggest hold but don't, e.g. when moving from vector spaces to modules. Typically these non-theorems are not the main point of the text but just warnings. If something like this is your intent, then this may be appropriate.
1 Well, except the fact that you said "Claim" rather than "Theorem" or "Conjecture" strongly suggests that you don't have much faith in it.
Best Answer
A result is the outcome of some process. As you indicated, mathematicians often call theorems results, since they are the outcomes of investigations. So, it's quite right to say something like There's is a result in complex variables to the effect that, etc.