Generally, exterior grade plywood contains phenol-formaldehyde that does not hydrolyse and give off harmful gas. Cotton and woollen components are often infested with mites, moths, or other insects and these can be deep-frozen to kill them. This technique can also be used for simple insect-infested wooden components. I would not recommend the deep-freezing of complicated jointed artefacts, such as the barrel of a barrel organ which may consist of several materials. The moisture content of the various timbers, paper, metal and leathers could expand and damage the object.
Leather is a different problem entirely.
Environmental control is the answer. Insect traps are readily available. They are intended to attract the insects to them and thereby establish if these insects are present, and if they are, attract them away from the organ. Corrosion in iron or lead can be active or inactive, and you need to be able to recognise the symptoms. Metals can be coated to prevent water vapour and oxygen from reaching the surface, but in many cases, iron is best treated with a solution of tannic acid, distilled water and ethanol.