I grill pizza on a regualr basis. (We have "Pizza Night" every week -- usually baked, but on hot summer nights we crank up the grill)
One thing I'd add though is to cook the dough by itself for a few minutes on one side (to firm but not browned) then flip it, add sauce and toppings and cook the rest of the way. Makes a nice crispy crust and avoids any problems with soggy interior.
A pizza stone (or plain, untreated stone tiles from Home Depot, etc) works great. Get the biggest one that'll fit (or an entire racks worth of tiles). You can leave it in the oven full-time -- it'll help even out the temperature in the oven when cooking other things too.
If you're cooking your pizza for 18 minutes, your oven isn't hot enough. Crank it ALL the way up. Most home ovens can hit 450-500°F. My pizza takes 10 minutes at 500, 12 at 450. Also -- preheat the oven and get it to temp with the pizza stone in the oven for at least 30 minutes before cooking.
+1 to the directions for the stone. My grandfather always turned his over to 500, put the pizza on parchment and when the parchment started to turn black the pizza was done.
when you grill the pizza do you use your stone?