Suppose the unit circle $\gamma$ in $R^2$ is not endowed with the canonical
inner product of $R^2$. Let the riemannian metric defined be
$g:=[2, 1;1, 1]$. So the length is measured with this metric on the
circle.
Q.1 Is there a way to construct the isometric embedding
of this $(\gamma,g)$ in some $(R^n,can)$?
Q.2 In general given any such $g$ can we always construct
an isometric embedding?



Whilst what you have given is not a Riemannian metric, the general question has a famous answer: http://en.wikipedia.org/wiki/Nash_Embedding_Theorem 

