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No, Bureau of Prisons officers are not authorized to carry firearms off duty in New York unless they have a valid concealed carry permit issued by the state of New York. The state's firearms laws apply to both federal and state law enforcement officers.
Yes. Under the Law Enforcement Officers Safety Act, any officer qualified to carry a concealed weapon in his home state may carry his weapon in any other state. Officers carry their badges and ID cards as proof of their right to carry a weapon.
It depends on the locality. Locally to me, the city officers carry a Sig 229M, county officers carry a Beretta 96D, and state troopers carry a S&W Sigma (I think...it may have changed). Each department sets its own regulations and policies for what its officers carry. Glocks are pretty common. I believe Glock holds about 60 percent of the police market.
Whether an auxiliary police officer can carry a weapon in other states depends on the laws of both the officer's home state and the state they are visiting. Generally, auxiliary police officers do not have the same legal authority as full-time law enforcement officers, and their ability to carry firearms may be restricted outside their jurisdiction. Some states have reciprocity agreements for concealed carry, but these typically apply to certified law enforcement officers. It's essential for auxiliary officers to research and comply with the specific regulations of the state they are in.
Sworn Illinois Police Officers are authorized by state statute to carry their weapon 24/7. In fact, many departments require that officers (while within their jurisdiction) carry their weapon and identification even while off-duty. Often, however, it becomes a matter of preferrence by the officer whether to carry off-duty or not.
C.
It depends on the State. In New York State, Agents or Officers of a SPCA are Full Time Peace Officers with Full Police Powers. In New York SPCA Officers carry, the same weapons as any police officer, included but not limited to handguns, long guns (patrol rifle), Taser, Batons, OC Spray.
state courts.
Most agencies encourage or require off-duty carrying of weapons by law enforcement officers. A recently-passed law allows off-duty, sworn and certified law-enforcement officers to carry concealed weapons any place in the U.S. (it is not prohibited by law to carry, but there may be restrictions on specific sites, such as court houses, churches, etc. that will impact the off-duty carrying out of the officers home state). In all states I am aware of, off-duty officers of that state may carry weapons anyplace they could carry them on-duty (which is everywhere).
There are 50 states, and 50 different answers. Game wardens are State officers.