A. I think this is a matter of opinion. Some people would be adamantly opposed to using wood unless the tree was grown “organically”.
Having a background in pesticides and the use of pesticides in orchards I would not be afraid of using wood from such an orchard provided it had been applied several months prior to it being cut. Most pesticides don’t have a particularly long life once they have been hit by the environment (sunlight, air, rain, etc.)
What residue is left is on the surface of the trees (bark) and not “inside” the tree. Most pesticides that end up inside the plant are called “systemic” and almost none are allowed for use in fruit and vegetable production. This is because it is thought that this pesticide would be moved around or translocated to the fruit and vegetables we eat.
If they were used they would be “translocated” through the layers of living wood just under the bark of the tree. This is the pathyway that “systemic” pesticides would take to allow it to move around inside the tree.
We have had similar questions about smoking wood where the bark has been “painted” with latex paint for sunburn protection. I have told people in the past when they had some concerns to remove or burn off the outside bark of the wood with some high temperatures and then turn down the temperature and put in your food for smoking.
If you are using wood with no bark on the outside then you don’t have to do anything. The “wood” of the tree is “dead” and an accumulation of the tissue which was alive previously and transported stuff inside the tree.