All paint is made up of the material that remains on the surface after the paint dries, and also a solvent that makes the material liquid so it can be applied to the surface with a brush or sprayer. It's the solvent that has "that paint smell." As the solvent evaporates, the smell becomes less and less, until it disappears completely when the paint is fully "cured." Until it's cured—which can take from days to months depending on the type of paint—the paint is slowly releasing fumes of solvent.
The toxicity of paint is indicated by the measurement of the "fumes" that come off of fresh paint, known collectively as "VOCs" (Volatile Organic Chemicals). These are gasses that are released into the air as the paint dries. Often paints contain a mix of various VOCs called "petroleum distillates," which may be any mix of solvents…whatever type has the best price on the day they are making the mix.
The US EPA has established an upper limit for VOCs in paint at 250 grams per liter (g/l) for flat finishes and 380 g/l for other finishes. Quite obviously the small amounts of paint used on the average model are much less than a Liter hence no warning labels
The paints with highest VOC levels are oil-based paints (also called solvent-based or alkyd paint). They use mineral spirits or other petroleum distillates as a base to carry the paint solids.
Water/alcohol based acrylic or "latex" paints are made from acrylic resins and various types of vinyl and use water/alcohol in place of a petroleum solvent, dramatically reducing the amount of VOCs.
OK, now the home-made filter box: First the activated carbon, a raw carbon containing material like wood, coal, peat, etc. is heated to a high temperature +600C in an inert atmosphere like nitrogen or argon. This carbonizes the material which is then exposed to oxygen or steam at high temperature. The "activated" carbon now has trillions of micro-pores giving it a vast surface area such that one gram can have a surface area of well over 1500 square meters. The exposed carbon atoms on these surfaces will attract and bind many types of organic molecules in effect storing them. The binding forces, van der Waals and London dispersion forces, are not very strong and the adsorbed gasses can be released back into the air, changes in temperature and humidity can accelerate this process. In addition since there are a finite number of binding sites the carbon can become "full" and will no longer accept more gasses.
The insulation material sounds to me like an attempt to mechanically filter out particulates of paint, i.e., overspray.
Now exhaust from his booth has to enter the box AND exit it at the same rate. Paint particulates and solvent gasses have to "flow"/pass through the insulation and activated charcoal. Simply blowing the exhaust over the insulation/charcoal will do nothing and it takes time for fumes to be adsorbed into the charcoal. Without knowing more this does not sound like a very effective "solution" to the fume problem.