I keep my egg covered for peace of mind. It keeps bird droppings off, water from penetrating to the inside and hopefully keeps the metal parts (bands/bolts, etc) from rusting. The exterior on my egg does get uncomfortable to touch, but has never gotten to the point of burning me. Thats with high heat searing/both vents fully opened or high heat cleaning.
The pizza's look fantastic! I'm a big fan of the San Marzano tomatoes. Not that it makes too much difference, but if they are true San Marzano's, they will have a paper DOP label on them.