I think pizza screens are best suited for doughs that contain a lot of sugar which, if baked on a hot pizza stone, could lead to excessive or premature browning--or even burning--of the bottom crust, often before the rest of the pizza has finished baking. Pizza screens are also a good choice for summertime baking where you would prefer not to heat up the oven for an hour or more to preheat a pizza stone to 500 degrees F or more. All that is necessary is to preheat the oven to the desired temperature, which can be as little as ten to twelve minutes. You should be able to bake just about any type of pizza on a screen with the exception of a deep-dish pizza, which requires its own pan or skillet, or a Sicilian or similar style pizza that requires a pan of some sort. I often use my screens in conjunction with a pizza stone to make pizzas that are larger than my pizza stone can accommodate. For example, I have started 16" and 18" pizzas on a screen and finished them on my stone, which cannot itself accommodate a pizza larger than 14". A pizza screen can also be used to lift a pizza off of a pizza stone by the thickness of the screen if the pizza is baking too quickly on the stone. Some people prefer the crispiness that can be achieved using a pizza stone as opposed to a screen, but for others that is not a big deal. Some say that they have been able to get crispy crusts using a screen. A screen is also a good choice for pizzas that are to be topped right to the outside edge and therefore not prone to falling off as they might if the pizza is loaded into the oven using a peel. It is rare to have a mishap when putting a pizza into the oven that is on a screen.
I always dress my pizzas directly on the screen. If the dough is highly hydrated or if the dough skin is very thin, I will usually pre-spray the screen with an oil spray, such as a canola oil spray, just to be on the safe side. I will also line up all of the ingredients and toppings in advance and move quickly to dress the pizza so that it doesn't have a chance to stick to the screen. Before using the screen for the first time, I suggest that you season it. The way to do this is discussed in this post: http://www.pizzamaking.com/forum/index.php/topic,2894.msg24858/topicseen.html#msg24858
I don't have a gas oven, so I cannot help you with how best to use your pizza screens with such an oven. In my electric oven, I have used screens at all levels, depending on the type of pizza I am trying to make or the oven thermodynamics I am trying to achieve.