Icemaker won't dispense Ice

I am actually having 2 problems. The first is that the ice maker is making ice but no ice will come from the door. When you push the button a noise can be heard but no ice comes out. The metal part in the ice tray does turn by hand but when we remove the tray the metal part in the back of the maker does not turn.

The second is a recurring problem. Occasionally the ice maker stops working and then water does not go to the ice maker buy falls over the back of the ice maker and down the back if the freezer, making a huge ice mess in the freezer but no actual ice. We remove the ice and clean it up and then the maker runs fine for a while, but a month or so later it happens again.

Thanks in advance for any advice for these frustrating problems!