Yes you are allowed to carry a gun on the Appalachian Trail as long as you have the proper licenses and permits for the gun in the states that go through the Appalachian Trail.
The Appalachian Trail people discourage the carrying of guns on the Appalachian Trail but some people do and it has not been a problem.
In some areas along the Appalachian Trail Hunting is allowed and in some areas along the Appalachian Trail hunting is not allowed.
There should be signs along the way that let you know whether or not hunting is allowed in those areas of the Appalachian Trail.
Hunting is allowed along 1,250 miles of the Appalachian Trail.
Hiking through the Appalachian Trail is actually free and if you want to just go hiking through the entire Appalachian Trail you can do so without spending any money.
But the Appalachian Trail does pass through some places and state parks that will require a fee for you to stay there overnight.
But you can sometimes camp in areas in a tent for free.
If you do want to spend time in state parks etc while on your journey through the entire Appalachian Trail it's recommended to have at least $1,500.00 per month on you.
The average spending per person hiking the Appalachian Trail is between $5,000.00 to $7,000.00
Hiking the entire Appalachian Trail is also a very difficult thing to do and some only hike halfway and then return.