Need help! Fairly simple questions.

Hey guys,

So I successfully proved that the sum of epsilons (the e in y=B0+B1*x+e) equaled 0.

Now how do I prove that the sum of xi's multiplied by epsilons also equals 0, AND that the sum of yi(hat)'s multiplied by epsilons equals 0?

Also, if I have an equation y=B0+B1*x+e where e are i.i.d. N[0,(sigma^2)/Wi)], what's an appropriate modifcation to the model so that the mean relationship remains the same?