Coil in top of freezer and top shelf in freezer have very large buildup of ice, melted it out but it soon froze back over

Do an old fashioned dethaw of your icemaker. Remove food in compartment, unplug or turn off freezer. Leave doors closed, and get a thick towel and a pot of boiling water. Place the towel under the pot, close the door, and let the heat from the pot warm up that frozen freezer vent. That will clear the ice so the fan can draw cold air out into the freezer

When you shut your top wire shelf you do a few things. First of all you activate your fan. Your fan sits in back of a large evaporator coil which is prone to defrost issues. I would imagine your fan is hitting ice due to a defrost issue. You may want to check your defrost system out. Ie your defrost timer.
or alternatively your fan could be getting old and needs replacing, it could just be making noise.
Joe B, Certified refrig.

There is a drain the back of the freezer that is frozen. Your trickle of water comes from the ice that melts off of the refrigeration coils during the defrost cycle. You need to turn off the freezer, then remove the rear cover from the back wall to expose the coils. With a hair dryer, melt all of the ice. You will find a drain under the coils. Melts all of the ice in this drain so water can get out of the freezer. When this drain is clear, you can put everything back together, and your freezer should work normally.

It sounds like it is not defrosting. Could be the heater, terminator, or the defrost timer. There is still ice built up behind the wall and won't take long for it to stop working again.

Make sure the fan in the freezer is running circulating the air.
How to check everything
http://www.acmehowto.com/howto/appliance/refrigerator/refrigerator.php

If the evaporator coils behind the back panel of the freezer are icing up because of auto defrost failure that will stop the circulation of cold air and eventually affect the freezer too.

check defrost timer, defrost heater, defrost thermostat. In most newer models the timer has been replaced by an electronic control board. If the heater and thermostat are ok it’ll be the control.

You can turn the defrost timer till it clicks and everything shuts down. The heater should now come on. If it does, replace the timer. If it doesn't, check the heater and defrost thermostat. Turn the timer again till everything starts back up to end the defrost cycle.

This happens when your thermistor gets frozen along with the coils or there is a clogging of the freezer drain which causes Ice to build up and spread to the system. Another issue is that your thermistor may be faulty, telling the system inaccurate temperature. Incorrect freon pressure can also be a cause. I would advise first to have the ice melted by letting the unit stay unplugged and have it cleaned while at it.

The excess frost in those areas is probably caused by leaky door gasket seals. They should be tight enough to hold a dollar bill securely all the way around. A defrost problem would start on the coils behind the back cover inside the freezer and would also appear on the back wall before it did on the door edges.

After you taken the back panel off and see a lot of ice forming on the coil before you use your hairdryer you should locate the timer and turn the center of the timer manually into defrost cycle. If you can see or hear the ice melting then your problem is the timer defective. Change the timer and that is all. But if it doesn't do anything then you will need a little bit more on electrical knowledge to deal with the rest on defrost temperature control and the element itself. Hope this help, God Bless.

The model number would help but it sounds like a sealed system leak or restriction. If thats the case I would not suggest trying this yourself. The water should freeze overnight and the frost should keep going to all the shelfs that have coils in them.

The reason for ice buildup is an air leak. Recheck all around the gasket by taking a dollar bill and closing the door on it and pulling it out. If you don't have resistance, you found your leak. Heat that area with a blow dryer to relax the gasket and close the door to let the magnet and seal set. Also, rub some vasoline on the hinge side gasket to keep it from dragging.