I can try to say a little about your first question, on derivations of $R$, *in characteristic 0*. As Mohan said, the derivations of $R$ are just derivations of the polynomial ring taking the defining equation to itself, but I'd add that it's a bit subtle what the collection of such derivations looks like. It's clear we have derivations of every positive degree $k$, for example by first applying the Euler operator and then multiplying by a polynomial of degree $k$. So, one natural question you can ask is whether there's a derivation of $R$ of negative degree. Note that such a derivation is locally nilpotent (though not all locally nilpotent derivations have negative degree).

The general context for your question is that we're given a ring $R=S(X,L):=\bigoplus H^0(X,L^m)$, which is naturally the section ring of a (smooth) projective variety with a chosen ample line bundle. By a theorem of J.M. Wahl ("A cohomological characterization of $\mathbb P^n$") if $R$ admits a derivation of negative degree, then $X$ must in fact be $\mathbb P^n$. Thus, in your example there will be no derivations of negative degree. (This crucially uses the characteristic 0 hypothesis.)

In fact, one can rule out differential operators of negative degree of any order on your $R$, not just derivations: one can show that the existence of a differential operator of order $m$ and degree $-e$ gives rise to an element of $H^0(X,(\mathrm{Sym}^m T_X)(-e))$, and in turn to an ample subsheaf of $\mathrm{Sym}^m T_X$. By powerful results of Miyaoka (Corollary 8.6 of "Deformations of a Morphism along a Foliation and Applications") this in turn forces $X$ to be uniruled (again, in characteristic 0!), which your K3 surface is not.