hi can I ask, when you set your fire up in the centre of the oven floor and you have lit it? At what point do you move it to the side of your oven? Is it when the fire is just burning nicely or is it when you've reached your correct oven temp?
By way of encouragement, what do you get? You don't need superhigh temperatures to cook pizza - 250C would do the job... after all, that's top temp for a conventional oven indoors. So, if it's a steady 250, have a go.
And... I don't have clean walls either: black as a hat. Cobblerdave put me onto the most likely answer to that, albeit a bit indirectly: in woodburning stove use, burning wood with any moisture content is blamed for build-up of soot and tarry deposits in a chimney. I think that despite my best intentions, I've been using wood that's not completely dry and I'm not at all sure any sort of 'cleaner' fire will see that dome soot-free now. It may also be that the dome itself was not completely dry when first fired, and may not even be now, after maybe 20 or 30 firings. I rendered it outside to waterproof it, and made it difficult to dry in the process.
I used to use straight-off-the-woodland-floor deadwood, progressed to stacked and stored woodburner logs split down extra small, and now use split-down pieces of oak and ash that I store in a crate that sits on top of a central heating boiler, drying.
Just yesterday, I checked some wood straight from the woodburner store (got a moisture meter for xmas) and it read 7-9% against zero for the stuff from the boiler top. Dave's trick is to make a mini-stack of selected bits at the doorway to the oven so they are drying while the fire's getting well established, then when you go to cook use the dried and warmed sticks to get instant blaze to cook the top of the pizza.
While I cooked pizza in my blackened oven at 250 using my dry wood yesterday, as an experiment I pushed some pieces of Ash freshly split down from full size into banana-size chunks into the fire, with the ends showing towards the door. Within minutes, water was bubbling and hissing out the end grain. Not sap - this wood's been dead and cut 2 years. It would be 30% plus if it was fresh cut.
I always got good fires - eventually - but it's a whole lot easier with really dry wood.
after a dozen big ish fires you should have driven any moisture out of your oven, as chas says dry wood is better and smokes less. try an upside down fire i use it every time and don't overload the fire, the oven will only absorb heat at a certain rate so any flames that may be licking up the chimney are just wasted heat, imo you're better going for a longer burn, if you're struggling to get the fires going a hair dryer works well and will get it flaming. at 500c you can't be far off clearing your dome though just keep burning and it will clear starting from the top first and working down to the floor.