Not really an answer, but instead
Edit: It seems appropriate to recall here Chevalley-Shephard-Todd's Theorem. It says the quotient of $\mathbb A^n_k$ by a remarkfinite linear group $G$ with order prime to the characteristic of $k$ is smooth (i.e. the algebra of invariants is polynomial) if and only if $G$ is generated by pseudo-reflections (codimension one fixed point set).
Once one localizes the problem at a point, as VA and Ben Webster did, this settles both 1 and 2 over arbitrary fields of characteristic zero. Of course VA argument is preferable as it is more direct/elementary.
Original answer. Below is my original answer commented by David Speyer below:
Over the category of smooth real manifolds the symmetric power of smooth curves is not smooth in general. The second symmetric power is a smooth surface with boundary and starting from the third symmetric power what we get are varieties with corners at the boundary.
Symmetric powers of smooth surfaces are still smooth as they are locally diffeomorphic to complex curves.
If nothing else these examples show that smoothness might mean different things
for algebraic geometers and differential geometers in algebraic geometry over $\mathbb R$ and in differential geometry.