I cured the oven and made a number of pizzas without the veneer. In fact I tarped it for the winter and did the veneer this spring. The only thing was I had to scrub the black residue off the bricks in order to stick the veneer on the front, so there was an extra step there that I wouldn’t have to do if I were more patient, but…I wanted pizzas
I definitely am going to seal it with the organic veneer sealer that @BrickWood recommends.
No, I have vertical brick walls and a roof covering the oven. I wanted the walls to be flush with the brick perimeter on the slab and a drainage plane if any water got past my exterior finish, so I left a 1/2” gap between the exterior brick and the stucco.
The only reason for waiting is if you want to avoid having to clean soot off the surfaces when you’re ready to veneer. Like @kgondoly I cured and did some cooking before winter. I plan to close my oven down in September to close off the front and do veneer.
“Water resistant membrane” is as simple as a good concrete deck/sidewalk paint. Yes, you must paint the stucco, especially if it is not veneered. It is composed of mortar, which is water permeable once cured and will allow water to soak your insulation, turn into steam, and eventually crack the stucco.
Do not put off this step, and plan to renew the paint annually unless you veneer later. When you do that, you will need to give the paint a good scrub with a brush and detergent to prepare it for veneering adhesive.