This mapping space is (I think) the same as the space of rationally connected components. In particular naive Mordell still works, since no two different points of a curve of genus $\ge 1$ are rationally connected!

The better analogy is not when you consider a general subgroup of $S_n$ but rather a "good" action of a group $G$ which acts on the $x_i$ by linear transformations (and therefore on the ring of polynomials by multiplicativity). You get a theory very much like symmetric polynomials (generated by certain elementary polynomials) if and only if $G$ is what's called a "complex reflection group" en.wikipedia.org/wiki/Complex_reflection_group. A more direct analogy with $S_n$ is "Coxeter groups" en.wikipedia.org/wiki/Coxeter_group

Well there is an alternative point of view you might like better, which is to consider the forgetful functor from augmented algebras to augmented vector spaces (vector spaces with a map to k). This functor also commutes with products and its adjoint is the ordinary free commutative algebra, $V\mapsto k[V]$ (with natural augmentation). I suspect this may answer your original question better, since this augmentation is precisely your list of $\lambda_i.$

I don't know if anyone has tried to formulate what an irreducible categorical representation would be, though highest weight representations do have a clear categorification (equivariant categories of sheaves on $G/U$).

$\rho$ is a function from elements of $G$ to matrices and $\rho(f)$ is defined as $\int_G f(g) dg$ (the "weighted action" by $f$).