What is $\mathbb{Q}_3(\sqrt{-6})^{\times}/\left(\mathbb{Q}_3(\sqrt{-6})^{\times}\right)^3$?

First, you needn’t have worried about what parameter you used: $\sqrt{-6}$ is just as good as $\sqrt3$. Indeed, if $\mathfrak o$ is a complete discrete valuation ring with fraction field $K$ and (additive) valuation $v:K^\times\to\Bbb Z$, and if$f(X)\in\mathfrak o[X]$ is an Eisenstein polynomial with a root $\alpha$, then $\alpha$ is a local parameter for the d.v.r. $\mathfrak o[\alpha]$. Since both $X^2+6$ and $X^2-3$ are Eisenstein for $\Bbb Z_3$, a root of either is good as a local parameter in $\Bbb Q_3(\sqrt{-6}\,)$.

Next, it may help for you to think of $K^\times/(K^\times)^3$ as $K^\times\otimes(\Bbb Z/3\Bbb Z)$. Whether or not, you were quite correct to see that all the contribution to $K^\times/(K^\times)^3$ comes from $1+\mathfrak m$. Here, of course, I’m using $K=\Bbb Q_3(\sqrt{-6}\,)$ and $\mathfrak m=\text{max}(\Bbb Z_3[\sqrt{-6}\,])=\sqrt{-6}\cdot\Bbb Z_3[\sqrt{-6}\,]$.

Now here’s something most useful: the multiplicative group $1+\mathfrak m$ is a $\Bbb Z_3$-module, via exponentiation. That is, for $z\in\Bbb Z_3$ and $\alpha\in\mathfrak m$, the expression $(1+\alpha)^z$ is well-defined, and all the rules that you know for $\Bbb Z$-exponents are valid. How’s it defined? Take any $3$-adically convergent sequence of positive integers with limit $z$, say $n_i\to z$. Then $\bigl\lbrace(1+\alpha)^{n_i}\bigr\rbrace$ is also $3$-adically convergent. I’ll leave it to you to prove that. Of course you see that the statement is true no matter what the $3$-adically complete local ring $\mathfrak o$ you’re dealing with. Note that the exponents are from $\Bbb Z_3$, nothing bigger.

Well: now that you know that $1+\mathfrak m$ is a $\Bbb Z_3$-module, what can you say about its structure? You know that it has no torsion, so it’s a free $\Bbb Z_3$-module. Of what rank? I think you can convince yourself pretty easily that the rank is equal to $[K:\Bbb Q_3]=2$; I’ll leave that to you, too.

Now it’s perfectly clear that $\bigl|(1+\mathfrak m)/(1+\mathfrak m)^3\bigr|=9$, the cardinality of a two-dimensional vector space over the field $\Bbb F_3$. Your enumeration of the elements is quite right, too.

Please don’t hesitate to ask for clarification or expansion of the above.