The answer to the first part follows from the law of the iterated expectation. See [here][1] for the proof. For the second part, strict exogenity $E[u_i|X]=0$ implies $E[u_i,X]=0$, but the reverse is not true. To see why this is, the former also implies non-linear (or any function of $X$) independence, e.g., $E[u_i,X^2]=0$.


  [1]: http://en.wikipedia.org/wiki/Law_of_total_expectation