Ryan, if you shoot me your e-mail, I will send you an excel spreadsheet that lists about 46 backcountry campsites that are very near the major streams in the park. It lists the trail you would use to get there, the distance, approximate elevation gain and estimated time to hike it with a pack (at least its an estimate by/for an old guy, you could probably hike there faster...)
Here is a glimpse of it:
My suggestion if you are coming through the Gatlinburg side of the park would be to go to the Sugarlands center, get a permit and go to backcountry campsite #24 above Elkmont. It is an "easy" 4.2 mile hike and puts you in the middle of some great water and scenery.
Campsite 24 is one of a handful of sites that require a reservation. Here's what you need to do:
If your itinerary includes a reserved site or any shelter, you must call the Backcountry Reservation Office at (865) 436-1231 to make reservations. The Backcountry Reservation Office is open from 8:00 a.m. until 6:00 p.m. daily. You may make reservations up to one month in advance of the first day of your trip.
Whenever folks ask for opinions about where to camp, #24 seems to surface more than just about any other.
As far as some of your other questions...Typically a backcountry campsite will have a limited number of sites that you can pitch a tent on. They are not all obvious and are sometimes a small distance away from each other. Other times that are all together. Most of the better pads will have a fire ring in place, so use one of them (don't create a new one). There will be rangers out an about. Its hard to tell when or where you will see one, but make sure you have your fishing license and a backcountry permit with you, and you will be fine. Also, make sure that you keep all of your food in ziploc (or better) bags and that you hoist your backpack up using the bear cables at each site. You clip your backpacks to the d ring and hoist it up in the air. Also make sure you don't have any food in your tent or on you, and wash well before going to sleep and you won't have any issues with bears. There are some other threads on here that talk about bears, but as long as you use common sense you will be fine. Over 8 million folks visit the park each year and only one person has ever been killed by a bear in the park. You are in much more danger of slipping and falling in the stream than being attacked by a bear...
That was probably more info than you wanted...