Why is i4 (E4) equal to zero? How can I arrive at that conclusion?
Since I5 is an empty wire, can I just say that the current there is
  the same as I2?


Yes it is a short circuit so I2 = I5


  But why isn't I4 0.4 and I5 0?


R4 is shorted by the short circuit hence all the current passes through that short circuit and none passes through R4. Therefore I5 is the same magnitude as I3 and I2.

If the I5 short wasn't there I4 would equal I2 = -I3