The third central moment of a sum of two independent random variables

Is it true that when X and Y are independent,

E ({X+Y}³) = E (X³)+E(Y³)?

This is just linearity of the expectation. You are assuming X and Y have expectation 0 and are independent. Develop (X+Y)^3, use linearity of E[.], then use independence and centrality to get E[X^2Y] = E[X^2]E[Y]=0 and E[XY^2] = E[X]E[Y^2]=0.