What could cause my washing machine to stop before the rinse cycle, it drains fine?
Most washers do not require the latch closed for most washing cycles. The one thing that does need for it to work is the spin cycle. So it sounds like the latch register the door closed is not working. I would try put the washer to the spin cycle and push in the tab or what not indicating the door/latch would be closed.

edit: I am a bit confused. It will spin if you move the timer to spin cycle? If so then it sounds like its the timer.