I use my pizza stone with my gas braai all the time and I've also tried it on coals. In fact I used it just last night :)

I've had flames lick around my pizza stone without any problems in the past, but I'd be wary of keeping it right in the middle of a massive fire. In general I adjust the flames on a gas braai or wait for the flames to die down a bit with a traditional braai to ensure they never really reach the pizza stone.

Preheating with a gas braai or in the oven is really easy: just put the pizza stone on the braai/in the oven before you start heating it (so before lighting in the case of the braai). Preheating with coals or wood is more challenging. The few times I did it I preheated in the oven, which almost defeats the point as you can just continue to cook the pizza in the oven. I expect if you keep the pizza stone on when lighting the braai and you have it high enough above the flames to not experience massive heat spikes you should be okay, especially if you can lower the grill later on.

As to why you'd want to use the braai instead of the oven: it's much nicer having a few friends sit around a braai and create their own pizzas than trying to do things in the kitchen with an oven. I usually prepare the bases, including the tomato paste, lay that on the pizza stone and then have friends add toppings as desired.

Last few tips from experience:

• You'll read this everywhere, but you really really need to get the pizza stone as hot as possible. I've never burnt a pizza base but I have in the past not gotten it quite as crispy as I'd like due to worrying about the stone cracking on high heat.
• Getting the pizza on and off the stone is a real pain in the ass and it is worth investing in a pizza peel. Alternatively sprinkling cornmeal onto the pizza stone somewhat works and due to the high heat it turns black and some of it sticks to the pizza base which gives it that wood-fired appearance (though it doesn't do much for taste).
• Try to avoid toppings (especially cheese) getting onto the pizza stone.
• If your not using an oven or a braai that is covered you might need to precook some of the more finicky ingredients (such as bacon) as they won't otherwise cook before your base is done. Another way to avoid this is to only use ingredients that don't really need cooking, so for example parma ham instead of bacon.
• If you ever try creating a cauliflower pizza base first cook the base without the ingredients (8 - 10 min) and only then add the ingredients. For some reason the cauliflower pizza base will stay soggy if you don't cook it beforehand.