= Solution
Take $f=g$ and $L=X_1+\cdots+X_n$. Permuting the variables produces another solution with the same linear term, so uniqueness gives
$$
F(X_1,\ldots,X_n)=F(X_{\sigma(1)},\ldots,X_{\sigma(n)}).
$$
For $a\in\mathcal O_K$, the one-variable case of part i gives a unique $\theta_a(X)\equiv aX\pmod{X^2}$ commuting with $f$. Applying uniqueness once more to the two ways of composing $\theta_a$ with $F$ gives
$$
\theta_a(F(X_1,\ldots,X_n))
=F(\theta_a(X_1),\ldots,\theta_a(X_n)).
$$
Thus $F$ is the addition law and the $\theta_a$ are scalar endomorphisms of the <Lubin–Tate formal group>.
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