For minimal surfaces a result of Dolgachev says that (possibly over the complex numbers only) that the image of $\mathrm{Aut}(X)$ in $\mathrm{Aut}(K_X^\perp)$ (the orthogonal complement of the canonical class) is a quotient of a subgroup of finite index of the full automorphism group of that lattice. Hence it is at least finitely generated. The normal subgroup by which one takes the quotient is the subgroup generated by reflections in nodal curves.

In the original (at least the version I have it says "regular locally Noetherian of dimension <= 1"). This excludes all Artinian rings except products of fields.

I meant the first sentence literally, I didn't understand. Now I do :-) Indeed, why we should get a subcomodule seems to be where the argument breaks down.

Notice that the Lefschetz hyperplane theorem works for any ample divisor. Hence, you do not need to know that $L$ has lots of sections in order to apply it.

Completing Dmitri's argument: By the Lefschetz hyperplane theorem (not the hard one...) $\mathrm{Pic}(X)=\mathrm{Pic}(D)=\mathbb Z$ and by the adjunction formula $(K_X+D)_{|D}=\mathcal{O}(-3)$ and hence $K_X=\mathcal{O}(-6)$ making $X$ a Fano variety of index $6$ which is not possible by a theorm of Fujita (Thm. 3.1.14 of Enc. of Mathematics, Algebraic Geometry V).

A comment to Dmitri's answer is given below but is too long to give as a comment. I write here as I believe that this will mean that he will be informed of my comments.

One way of getting this isomorphism is that it is a special case of the Iwasawa decomposition for a general semi-simple (real) Lie group. This allows us to get many specific references, Helgason for instance.