Yes, that's always the way you find a conjugate- replace i by -i.

As for your "zz* = (Re z)^2 - (Im z)^2", that is correct if by "Im z" you are including the "i". That is, if z= a+ bi, Re z= a but some texts use Im z to mean just b, others bi. With the understanding that Re(a+bi)= bi, then, yes, zz*= a^2- (bi)^2= a^2+ b^2.