Basic pizza instructions (not recipe)

Follow the instructions for the dough that you purchased.  Parchment is the way to go and you should be fine at this temp. Premade dough often contains sugar and oil so 550 may be too hot for this dough, thats why it is better to follow the instructions.  Make sure to let your stone preheat for about 30 minutes.

Parchment paper, according to this site: http://www.culinarylore.com/food-science:is-parchment-paper-flammable will begin to smoke around 420f.

There's silicone in parchment paper so it's not likely to burn like paper, but on an egg with live fire nearby - it will flame if it gets a chance, so make sure there's nothing hanging over the side of the pizza stone.

I've used parchment paper in my egg, but only at 300f or less, and always with something wet inside. (Fish, veggies, etc.)

High temp pizza will likely not stick with a little cornmeal on the stone.

cooking time depends not only on temp but thickness of the dough and where you are cooking in the dome.  A thin pizza I will cook hotter.  Thicker pie will need more time for the dough to cook through and I don't want the toppings to burn, so lower temp.  Ballpark time for me is around 12 minutes but I don't have an exact answer.

I usually don't use parchment.  I roll the dough out on a King Arthur silicone rolling mat.  Then get the peel covered with cornmeal.  get the dough onto the peel, stretch a little, and make sure it slides easily.  Assemble and pop onto the egg.  You don't have a lot of time here, and make sure the pie slides before you try transferring to the egg.

that said, this winter, when i was cooking pizzas in the freezing cold snowy weather, doing it on parchment was super easy.  Just slide it out after 5 minutes, rotating the pie in the process if needed, and finish cooking.  It gets singed around the edges but so far I haven't ignited it.

How long do you cook pizza for at 550'?  
  -- The time to cook a pizza depends on the temp, dough thickness, dough recipe and personal preference as to doneness. Ball park at 550ºF is 10 minutes plus or minus quite a bit. If you are buying pre-made dough, check the label as to the recommended temp to cook at. You can up the temp some, but don't go too far.

Do I need to let the dough get to room temp (I am getting remade from food store?)
  --  Yes. Let the dough rest for a couple of hours before stretching.

And, people say to cook it on parchment paper on top of the stone the whole time for easy cleanup - won't the parchment paper burn/get weird in the hot egg?
  --  Only the exposed portion will burn. As long as the ash/burned paper doesn't land on the pizza, it will be ok. I pull the parchment after the dough has set to prevent this (about 2 minutes into the cook). If I am making only one pizza, I will use semolina. When making multiple pizzas, I will use parchment.