Mass murderer: I generally avoid making blanket statements, because there is usually an exception to every rule. BUT, generally speaking, I favor capital punishment for mass murderers. They won’t need a gun in hell.
Mentally deranged person: No, mentally deranged people should not have access to firearms, or knives, or clubs, or cars, or any other potential weapon. They won’t in an inpatient mental facility.
6-year-old: As long as he’s supervised by a competent adult, I have no problem with a 6-year-old learning to shoot a gun, assuming he’s not an idiot. I would leave that up to the discretion of the parents. Presumably, he won’t be living on his own, so he can “own” the gun under Mommy and Daddy’s roof.
Illegal 18-year-old male: No. Since someone who entered the country illegally isn’t supposed to be here in the first place, he doesn’t have a right under federal law to own or possess a firearm, and I support that aspect of the law. Not sure why stating an intent to become a citizen would be relevant. He can get the gun when his status changes.
“To keep and bear arms”: Yeah, I support that, and you’re probably asking the wrong guy, since I put the threshold pretty high and live in a state (Mississippi) where I could legally own a bazooka or a fully-automatic heavy machine gun such as a .50 cal. Browning.

The thing is these weapons are expensive thanks to limited (legal) supply, and I simply have no desire to own one.