Yes, mostly for their own protection
But, it is only legal if the person otherwise qualifies to carry a firearm. EMTs (assuming you mean Emergency Medical Tech) are not normally sworn law enforcement officers, therefore, they cannot legally carry a weapon just because they are EMTs. They must carry in accordance with state law, whatever that is. However, I doubt any city in the U.S. allows its EMTs carry firearms. I know that MOST cities don't.
No
No. Cannot carry, touch, shoot, or have access to firearms.
No. Most correctional officers do not carry firearms.
Yes
Each branch of the military and each branch of many federal agencies carry firearms and train with firearms. The IRS has agents that carry firearms. Probably the largest users would be the Marines and Army.
The same authority that enables them to carry firearms.
to beat people up
Depends on the laws of the jurisdiction which you are in.
because they like to teach
NO
You need to ask a lawyer who is familiar with firearms law.
No, Oregon has no law requiring one to register their firearms in any way. The purchase of firearms requires a background check that takes 10 min. Other than that, the only registration/license you need is a concealed carry permit if you choose to carry a concealed weapon. You are allowed to open carry in this state as well.