Question

1. In Artin1968 he considers $\underline{analytic}$ equations, but over the ring $R=k\{x_1,..,x_n\}$. In Artin1969 he works with $R=k\{x_1,..,x_n\}/I$, not necessarily regular, but considers $\underline{polynomial}$ equations.
   Is there some version like this: "Let $R$ be a local Noetherian Henselian ring(not necessarily regular), over a normed field. Given an arbitrary (possibly countable) system of analytic equations over $R$, with a solution over the completion of $R$, there exists also a solution in $R$, sufficiently close to the formal solution" ??

2. What is known for non-Noetherian rings? e.g. for $C^\infty$, $C^r$?
   (Actually, for $C^\infty$ I learned about one approximation theorem, unpublished in the old USSR times..)

Answer 1

Concerning (2), here are some references:

For certain subrings of $R[[T_1,\dots,T_N]]$ where $R$ is a complete valuation ring of rank 1, see:
H. Schoutens: Approximation properties for some non-Noetherian local rings. Pac. J. Math. 131(2), 331–359 (1988).

For any henselian valuation ring, with fraction field $K$, such that the completion $\widehat{K}$ is separable over $K$, see my paper:
An extension of Greenberg’s theorem to general valuation rings, Manuscripta Math. 139, 153–166 (2012).

The case of a henselian valuation ring of rank 1 is already mentioned in Elkik's thesis:
R. Elkik: Solutions d’équations à coefficients dans un anneau hensélien. Ann. Sci. École Norm. Sup. (4) 6, 553–603 (1974) (see Remarque 2, p. 587),
and treated in more detail in chapter 1 of
A. Abbes: Éléments de géométrie rigide I. Progress in Mathematics. Birkhäuser, Boston (2011).

For rings of differentiable functions, perhaps you should look at Tougeron's papers.