If you use the correct ammo, it does not permanently damage the animal and they will soon learn to stay away from your yard. Get one cat and keep it calm and happy. Use the smelliest aerosol can you can find,of say deoderant and spray close to the cats ear BUT not on the cat. Now spray the area the cat poos in. When the cat goes to this area it will smell the aerosol and relate this to the time you sprayed it near its ear.
Cats hate the hissing noise near their ears.
You will have to do followup sprays if the cat again shows interest in the garden. In other words you are training the cat to stay out. Sling shots, bricks etc will not stop an animal from doing whatever it likes. All you are teaching the animal to watch out for people with slingshots because the animal will not associate the offending behaviour with the slingshot. The act of doing wrong is what must punish the animal, this is basic animal training.