How do you disinfect this kinda thing? Neighbour gave us a set but it's old and dusty. Do you wipe the entire thing with disinfectant?

You can dismantle that in parts right, you can take it to the shower piece by piece and wash with soap. Try those eco or organic detergents, otherwise warm water and regular soap will do. You may want to soak the balls by batches into a small basin or in the tub.