It is perfectly safe to use a coal, wood or gas fired BBQ or fireplace beneath the louvered roof. In fact it is extremely common for outdoor kitchens to be installed beneath the structure. The only stipulation is that any flues from for example pizza ovens be positioned a minimum of 0.3m from the underside of the roof louvers and that if the structure is fitted with side screens any heat surfaces are kept a minimum of 0.5m from the screen fabrics.
Additionally with the roof louvers open at 10 degrees, this will allow for ventilation from heat build up and smoke beneath whilst still offering rain shower protection from downwards rain.