Are there relations in this ring which are not consequences of the fundamental relation (1)?.
If by this ring you mean the subring of Maps($C^\infty(\mathbb{R}^n),C^\infty(\mathbb{R}^n)$) generated by partial derivatives, then the answer is, no. Indeed, the map from $\mathbb{Z}[v_1,\ldots,v_n]$ (the free commutative ring in $n$ generators, aka polynomials with integer coefficients), to Maps($C^\infty(\mathbb{R}^n),C^\infty(\mathbb{R}^n)$), sending each generator $v_i$ to the partial derivative operator $D_i$ is injective, since from
$\sum_{\alpha\in\mathbb{N}^n} c_\alpha D^\alpha (f)=0$ for all functions $f\in C^\infty(\mathbb{R}^n)$,
we can conclude that all coefficients $c_\alpha\in \mathbb{Z}$ are zero, by applying it to monomial functions $f(x_1,\ldots,x_n)=x^\alpha$. (Here $\alpha =(\alpha_1,\ldots,\alpha_n)$ is a multi-index and $x^\alpha = x_1^{\alpha_1}\cdots x_n^{\alpha_n}$ etc.)