The ovens are pretty much self-cleaning.
Just like the oven in your home, (but at much hotter temps), you simply get your oven to 800+ degrees and spread the fire & coals all over the hearth cooking surface. This will pretty much make any topping or dripping on the firebrick disappear. Think Arnold Schwarzenegger at the end of Terminator 2.
After that, let the oven cool down a bit and put all the ash in a galvanized trash can w/ a bit of water the bottom. Soak and old t-shirt in water, wring it out then wrap the t-shirt around your brush / broom. Then “mop” the firebrick surface but don’t try to soak the brick - more steam-cleaning really.
Now, you mentioned that the rain sometimes enters the oven during a hard rain. Plz make sure that you don’t choose this time to clean out the the oven w/ a large fire. Wet brick and hot fire do not mix. The heat will turn the water into steam and steam will crack the firebrick.
Try to keep your oven dry - make sure you keep your oven covered w/ a tarp.