Right now the handle on the toilet tank is facing towards the back. Is there a way to make it face the front?

Not likely.

When you push down on the flush handle the rod turns counterclockwise. If you could make a handle that faced forward the rod would turn clockwise and not work. Try loosening the nut inside the tank that hold the handle, once loose try to rotate the entire mechanism, handle and rod,and see if it will work for you.

If he does that, won't he have to pull the handle upwards to flush?

No. First if you turn it upside down (180 degrees in the hole) the rod would not bent properly. If you re-bend the rod so that it reached the flapper, the handle itself would still be up side down.