How are proofs formatted when the answer is a counterexample? Suppose it is asked:

Prove or find a counterexample: the sum of two integers is odd

The fact that 1 + 1 = 2 is a counterexample that disproves that statement. What is the proper format in which to write this? I will provide my attempt.

Theorem: the sum of two integers is odd
Proof:
1 + 1 = 2
=> the sum of two integers may be even

I understand the math, but I'm not sure how to properly present a counterexample relative to an initial theorem. Was this an acceptable presentation of a proof?

The "theorem" you are proving, in the specific case you give, is not "the sum of two odd integers is always odd." A "theorem", by definition, is a true (or at least provable) statement. So the "theorem" cannot be "the sum of two odd integers is always odd," because your argument shows that statement to be false.
In fact what you are proving is "the sum of two odd integers may be even." So you could format it as follows, which would be better.

Theorem. The sum of two odd integers may be even.
Proof: The number $1$ is an odd integer, and $1+1=2,$ and $2$ is an even integer.

However, this statement is not really deserving of the name "theorem;" the word "theorem" means "behold" and is usually reserved for results that are worthy of beholding. Better would be "proposition," or, in my opinion, better still would just be "claim."

Side note: You are almost certainly aware of this, but we can of course do better in this situation. The sum of any two odd integers will always be even.

Proposition. Let $m$ and $n$ be odd integers. Then $m+n$ is even.
Proof: Since $m$ and $n$ are odd, there exist integers $k$ and $l$ with $m=2k+1$ and $n=2l+1.$ Hence $m+n=(2k+1)+(2l+1)=2(k+l+1).$ Since $k$ and $l$ are integers, so is $k+l+1,$ and it follows that $m+n$ is even, since it is an integer multiple of $2$.

