For a) the answer is no. E[X1*X2] is not the same as E[X1|X2]. E[X1X2] = Integral over region for (X1,X2) x1x2*f(x1,x2)dx1dx2 whereas E[X1|X2=x2] = Integral x1*f(x1|X2=x2)dx1.
X1 and X2 are independent if for all values of X1 and X2 you have the joint distribution to be separable. In other words:
P(X1 = x1, X2 = x2) = P(X1 = x2)P(X2 = x2) for all x1 and x2.
To compute conditional expectations you need to find the conditional distribution. To do this use the relationship P(A|B) = P(A and B)/P(B) and then integrate with respect to the mean of A.
To get you started on the table exercise, note that
E[X1X2] = -1*-1*P(X1=-1,X2=-1) + 0*-1*P(X1=0,X2=-1) + ... + 1*1*P(X1=1,X2=1).