Let $W(t)$ be Brownian Motion. Find the mean and variance of $\int_o^t W(s)dW(s)$ and $\int_0^tW(s)ds$.
I think I would set $X(s) = \int_o^t W(s)dW(s)$ as a random variable, take $\frac{d}{ds} X(s)$ and find the mean and variance from there but I am not sure that is correct.
Three of your four questions are easy. First, one can interchange mean with ordinary integration (using Fubini's theorem), which gives zero for the mean of $\int_0^t W(s) ds$. Second, one can use Ito's formula to prove that $\int_0^t W(s) dW(s) = \frac{W(t)^2-t}{2}$, from which one can read off the mean and variance straightforwardly. (Alternately one can use general properties of stochastic integration, such as the Ito iometry, but that will only help you get two moments. My suggestion allows you to get whatever moments you want.)
The last question, which amounts to computing $E \left [ \left ( \int_0^t W(s) ds \right )^2 \right ]$ is harder. Why not give us an attempt on that one?