How to install a manufactured stone surround on a zero clearance fireplace

Drywall is out of the question due to weight, plywood might not give you the proper adhesion.

I would start with cement board, then attach the metal lath to that. After the metal lath is secure, you'll have to apply a scratch coat of a sand and cement mix before applying the stone.

Here is a good guide for applying stone veneer.

Did the transparent glass "back wall" explode because you added the front side glass doors and as a result the temperature in the fire box area went way up???

You need to consider carefully the airflow and draw requirements of the fireplace and the chimney that vents it out through the roof. Changing the design can really change the temperatures of the internal space!!

I would think that if you were to block off the far side with bricks that you may need to line the inside of the brick with special heat resistant fire bricks. Regular bricks could get extra hot and shatter too.

If you really need to have the front side glass doors make sure that there is a part of the design that permits the proper amount of air flow in around the sides of the door assembly for the chimney draw to work correctly.