Are all metric tensors diagonal?

Being diagonal is a coordinate-dependent concept: the components of the matrix associated to the metric tensor depend on the coordinate system you use. Thus a very simple example of a non-diagonal metric is the standard Euclidean metric $\delta = dx^2 + dy^2$ on $\mathbb R^2$ in the coordinate system $(x,z) = (x, x+y)$, where it has the coordinate expression $$\delta = dx^2 + d(z-x)^2 = 2dx^2 + dz^2 - 2 dx dz.$$

No, in fact, there's some very famous solutions that have non-diagonal metrics. Such as the Kerr metric for a rotating black hole in General relativity.