It's great to cook outside and although some may argue that it is not exactly sustainable, it does have great heath benefits, in being outside connected with nature and the social aspect of a get together.
We try to make use of all the heat for example using the charcoal burner to boil a kettle, even before its used in the portable BBQ. We also boil it on the pioneer stove whenever that is alight.
We try to use windfall wood to do most of our cooking. Windfall is old wood that has fallen from the trees naturally, it is easy to snap and quite often has moss on it.
There is an argument that burning deadwood is actually better than allowing it to rot which will produce methane and carbon dioxide.
If your source of wood involves cutting live trees then it is only carbon neutral if you know that the rate of cutting does not exceed the long-term regrowth.
It is really important that 'green' wood, sticks that bend rather than snap isn't used. You also do need to be careful as most wood belongs to someone and this could be classed as theft. The odd handful and not a car full is fine or get permission from the land owner.
We have four main types of burners. This might seem excessive, they are all good quality and have been used for years
A tripod with with a large cooking pot
A large kettle BBQ can be used like an oven
A small portable BBQ
A pioneer stove that we use inside our tent.