There is an old paper of mine called "On the Differentiability of Isometries" in which I show that if you know a Riemannian manifold $M$ only as a metric space, i.e., you just know its point set and the metric function $d(x,y)$, then from that information you can recover first the differentiable structure and then the Riemannian metric tensor (i.e., the inner product on each tangent space). I think that answers your first question. The paper is downloadable from here:
http://www.ams.org/journals/proc/1957.../S0002-9939-1957-0088000-X.pdf
http://www.ams.org/proc/1957-008-04/S0002-9939-1957-0088000-X/S0002-9939-1957-0088000-X.pdf
