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(Actually, for $C^\infty$ I learned about one approximation theorem, unpublished in the old USSR times..)



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: For any henselian valuation ring, with fraction field $K$, such that the completion $\widehat{K}$ is separable over $K$, see my paper: The case of a henselian valuation ring of rank 1 is already mentioned in Elkik's thesis: For rings of differentiable functions, perhaps you should look at Tougeron's papers. 

