What to use on top of a baking stone: parchment paper, a silicone sheet or a thin steel tray?

Ideally you'd use nothing. Direct contact with the hot stone delivers the heat to the food best. Spilt food on the pizza stone can be scraped off when cold and any residue will be sterilised when the stone is preheated, as it should be for some time. It's best to leave the stone in the oven after removing the pizza (or put it back in) as the considerable residual heat will help char the residue to a loose charcoal/ash mix that can be brushed off, leaving what's effectively a seasoned surface.

Although I store mine in the oven, I usually take it out when I'm using the oven and not the stone, as it's 5kg of granite and that takes a lot of preheating, slowing my cooking and increasing energy consumption. If I'm not planning to use it for a while, or haven't used it for a while, I might leave it in while baking bread, being sure to preheat generously. You can also bake off residue by putting the already-hot stone under the grill (broiler). This can get a bit smoky though.

Having said all that, I have been known to use non-stick cooking liner, when I want to cook something on the stone but don't want to risk imparting a flavour to it (or picking up a flavour from it), or when cooking a really sticky dough like naan.

FYI, I use my baking parchment multiple times—until it crumbles in my hands. I’ve never used it on my pizza stone, tho. As for cleanliness, my pizza stone is almost black (I figure from grease and oil in what I bake on it) and things that bake on to it (cheese, etc) scrape off with a plastic scraper I have.