Problem #4 - No agitation or spin

This is an indication of a worn or broken drive belt or in the case of direct drive washing machines a worn direct drive coupler. If the belts look solid they may still be slipping due to glaze. Another possibility is that the motor needs to be replaced. 

Problem #5 - Water will not empty

This is caused by a malfunctioning pump. Check the drive belts on the pump and if they seem to be OK try to move the belt by hand (with the machine disconnected!) If it seems to be stiff the pump needs to be replaced.

If the pump and belts seem to be OK check the drain hose for obstructions. 

Problem #6 - Washer shakes during the spin cycle 

Washing machines need to be leveled carefully to avoid this problem. There are usually adjustable feet on each corner of the machine for adjust the level. Make sure the machine is installed on a sturdy floor or reinforce the underfloor if necessary. 

If the machine is perfectly leveled and still vibrates during the spin cycle check to see if the damper pads are worn. 

Note that some amount of vibration is normal and that front loading machines are more prone to this problem than top-loaders. 

Problem #7 - Washing machine damages clothing 

This can happen if there is not enough water during the wash cycle. Be sure to use the recommended water level for the load of laundry. Open the lid during the wash cycle to make sure the clothes are circulating freely and if not, stop the cycle and add more water. 

This could also be caused by clothes getting trapped under the agitator. Check for rough spots around the bottom of the agitator which could be catching the clothing.