That would be a good idea. The principle of radiocarbon dating is that when a plant dies or the branch of a tree is cut, it stops taking in CO2 from the air, which includes a certain proportion of ¹⁴C, i.e. atmospheric carbon has a certain fixed ratio of ¹⁴C :¹²C, the latter being the normal, stable isotope of carbon. Over time, then, the ¹⁴C, being unstable, decays. So the ¹⁴C :¹²C ratio of a wooden or plant-derived object will decline in a predictable way with time. So you analyse the object to determine the ratio and that tells you how long ago it ceased to be part of a living plant.
Charcoal is made by partial burning of wood, either as a byproduct of a wood fire or by design, in a fire that is covered over to control the combustion so that as much as possible of the wood is turned to charcoal. Early people making charcoal drawings probably used pieces from an open fire, as there would always be some of that to hand.
More generally, since you are not a scientist, it would be prudent for you to check some of these things before you try to dismiss them.