Why does elevator still raise and lower the nose when the aircraft is steeply banked? In a steep turn, the aircraft is basically sideways. However, when executing a steep turn, one still raises and lowers the nose of the aircraft by using the elevator. 

Intuitively, it seems to be that in an extreme bank, the elevator and rudder have essentially changed roles, since they have been rotated with respect to their normal orientation. It seems to me that in a steep turn, pulling back on the yoke should tighten the turn. But this seems to not be the case in practice, since getting the nose up in a steep turn still involved just pulling back on the yoke and/or trim. 

Why is this the case?
At 45 degrees Angle of Bank pulling back will both tighten the turn, and cause the nose to rise equally because you are halfway to 90 degrees.  You are correct though, at much steeper angles of bank the control inputs function the other way:  In a very steep high G turn the angle of bank is used to control altitude, and back pressure is used to control the rate of turn.  You definitely don't want to use rudder in this case though, bad things can happen in an accelerated stall with yaw.