Yes you can take your dog with you on the Appalachian Trail but the dog must be on a leash and must be under your control at all times.
You cannot let the dog hike freely without being on a leash on the Appalachian Trail but the dog can be with you.
Just note that while dogs are allowed on the Appalachian Trail the dogs are not allowed in some of the state parks and other locations along the Appalachian Trail.
There are no permits required to hike the Appalachian Trail or just go for a walk along the Appalachian Trail.
Some people take a brisk walk for a mile or two or less and not do any actual hiking and even if you're not into hiking it's a good thing to walk the Appalachian Trail at least for a little while and then you can turn around and go back where you started.
You don't need any permits or need to pay any fees just to hike the Appalachian Trail.
However some state parks or campsites along the way may require permits or fees to stay at hose campgrounds or state parks.
The way that some people bathe on the Appalachian Trail is when they stay in motels or hostels along the way of the Appalachian Trail.
Some people also just do something like a sponge bath and may use water from a creek to bathe and stay clean.
You can stay pretty clean by just doing a quick sponge bath to wash away some of the sweat or even yourself a solar shower that is for camping and hiking.
You fill the solar shower with water and the heat from the sun heats the water up so that you have a nice warm shower when ready.
Some people camp in tents along the way and some people stay in some hostels or hotels in towns along the way of the Appalachian Trail.