Yes.
Some departments will specify what model the carry weapon is.
The laws on this vary by state.
A police officer
In general, yes, under federal law HR218, a retired officer can carry a gun in any U.S. state.ADDED: HR 218 requires that the retired officer re-qualify with his sidearm once per year to the standards of the agency fom which he he retired or that of his local law enforcement agency.
need state to see if it's open carry
In states where concealed weapons permits are issued, private investigators can carry weapons (if they have the permit). Also, in some states if you are a retired or active peace officer you can carry a gun with the proper endorsement.
Federal law provides that a peace officer can carry a firearm in any state, and retired peace officers that meet certain requirements (mainly weapons qualification) may do so as well. NYC is trying not to comply with the Federal alw on this.
On Halloween, yes, but in real life, no. Too young to carry a gun. Child labor laws prohibit him from working as a policeman.
Yes. They are licensed police officers just any anyother cop. Federal Law gives them the right to carry off duty. See HR 218
It depends on what kind of school. See the related link for more information.
Generally, no, especially not for K-12.