Edit: after the edits of OP this seems to be an incomplete answer.
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 (where I consider composition as multiplication), 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.)