Does a second moment of zero imply a mean of zero? I was wondering about this. I have a second moment that is zero. Can I conclude that if $E[x^2]=0$ then $Var[x]=E[x^2]-E[x]E[x]$ implies $E[x]=0$ since the variance must be non-negative? (If not, can I imply anything else from that, in particular, regarding if we also have mean square convergence?) 

The answer to your question is yes.
Alternatively, you can use Hölder's inequality and obtain the following inequality
$$
0\le|\operatorname EX|\le\operatorname E|X|\le(\operatorname E|X|^2)^{1/2}.
$$
