Washers will fill under any water temperature and do agitate and spin, then will stop while filling during the rinse cycle. You can lift the lid and close it, and it will begin filling, slowly, again. You need to lift and close the lid 3 or 4 times for it to fill completely, then it will finish a full cycle. Any idea what is making it stop during the water filling?
Darn, since it works through the was cycle there is nothing mechanically wrong with the machine.
But this model has both a timer and an electronic control, and probably one of these is bad but you will have to try replacing both to see which one you need.
If you buy from this site you can return parts you do not need or parts that do not fix the problem.