You can shoot pretty much anything that will fit in the pouch with a slingshot. Different sizes and weights are going to behave differently. I used to use 11mm paintballs.
Check state code. Generally, you won't be disqualified from gun ownership unless you've been involuntarily committed or receive SSDI for certain mental disorders. For now at least, background checks just cover legal history and the state doesn't have access to your medical history.