Does equality in distribution imply equality of expected value?

Yes, this is true.

The expectation value depends only on the measure $P^X$ on $\mathbb R$ induced by the random variable $X$, and $X$ and $Y$ are, by definition, equal in distribution if and only if $P^X=P^Y$.

You can find this here: http://en.wikipedia.org/wiki/Expectation_value