1. When you write $x' = sin x$ they don't have same domain so the equivalence of the metrics will only be for a certain coordinate patch (I think)
2. When you do the coordinate transformation $x' \to \lambda x'$ will have a different range than the original. For example if I limit my metric to $ds^2 = dx^2 + dy^2 $ to $(-1,1)$ and transform the metric to $\lamda x$ and $\lambda y$ then my range has contracted to $(-1/lambda,1/lambda)$ in this case the metric has not rescaled since the domain is different