Backpropagation: In second-order methods, would ReLU derivative be 0? and what its effect on training?
Yes the ReLU second order derivative is 0. Technically, neither $\frac{dy}{dx}$ nor $\frac{d^2y}{dx^2}$ are defined at $x=0$, but we ignore that - in practice an exact $x=0$ is rare and not especially meaningful, so this is not a problem. Newton's method does not work on the ReLU transfer function because it has no stationary points. It also doesn't work meaningfully on most other common transfer functions though - they cannot be minimised or maximised for finite inputs.

When you combine multiple ReLU functions with layers of matrix multiplications in a structure such as a neural network, and wish to minimise an objective function, the picture is more complicated. This combination does have stationary points. Even a single ReLU neuron and a mean square error objective will have different enough behaviour such that the second-order derivative of the single weight will vary and is not guaranteed to be 0.

Nonlinearities when multiple layers combine is what creates a more interesting optimisation surface. This also means that it is harder to calculate useful second-order partial derivatives (or Hessian matrix), it is not just a matter of taking second order derivatives of the transfer functions. 

The fact that $\frac{d^2y}{dx^2} = 0$ for the transfer function will make some terms zero in the matrix (for the second order effect from same neuron activation), but the majority of terms in the Hessian are of the form $\frac{\partial^2E}{\partial x_i\partial x_j}$ where E is the objective and $x_i$, $x_j$ are different parameters of the neural network. A fully-realised Hessian matrix will have $N^2$ terms where $N$ is number of parameters - with large neural networks having upwards of 1 million parameters, even with a simple calculation process and many terms being 0 (e.g. w.r.t. 2 weights in same layer) this may not be feasible to compute.

There are techniques to estimate effects of second-order derivatives used in some neural network optimisers. RMSProp can be viewed as roughly estimating second-order effects, for example. The "Hessian-free" optimisers more explicitly calculate the impact of this matrix.