Camping stoves are generally not safe for indoor use. They can produce fumes or carbon monoxide that would be fine outdoors, but dangerous indoors. Check the warning labels before buying anything to use inside.
As for the actual cooking, I don't think that you'll get the results that you expect.
My experience includes an MSR Whisperlite International backpacking stove (white gas), and Coleman two-burner stoves in both white gas and propane variants. Anecdotally, the backpacking stove has limited control, while the two-burner stoves don't quite have the oomph of a real gas stove.
The Coleman links that I provided indicate that the white gas stove has burners that put out 7,500 and 6,500 BTU, while the propane stove puts out 10,000 btu on both burners. Using the REI-provided time for boiling water, I calculated that the backpacking stove puts out about 4,500 BTU (and other backpacking stoves indicate similar times, regardless of fuel).
By comparison, my consumer-grade natural gas-powered kitchen stove has two burners that put out 15,500 BTU, one that puts out 9,500, and one (the simmer burner) that puts out 5,000. Viking offers normal burners up to 18,000 BTU, and a wok burner of 27,000.
However, white gas stoves aren't okay indoors. They're fine while they're burning, but lighting them is a bit dramatic, and once you shut them off they'll put out half-burned fumes for several minutes.
Similarly, propane camping stoves usually put off too much carbon monoxide to be safe indoors.
The stone will be fine kept in your oven all the time, but there are trade-offs. The biggie is that your oven will preheat slower than you're currently used to--potentially quite a bit if you get a large or heavy stone. And you will really need to make sure that the stone has reached the desired oven temperature (by preheating longer) or the way things cook in your oven will be thrown off. The oven may claim to be fully preheated before the stone is really up to temperature because the thermostat just measures the air temperature, which can be higher than the stone, or even the sides of the oven for a while.
The upside is that your oven will retain heat better and generally be more even in its heat. So if you are willing to properly preheat that stone, you'll probably benefit in the long run.
As you've divined already, you'll have to put it on one of your oven racks as low as it will go toward the bottom of the oven. You don't want to put it right on the coils, but you want it as close as it will go. This means you'll have one less rack available to you. In practice this may not matter much, but it's a thing.
I wouldn't recommend putting a pan directly on the stone, but that's because it would completely change how heat gets to the pan. When a pan is on a rack, there's heated air circulating around from all sides. If it sits on a preheated stone the pan now has direct heat conducted to the bottom--which will change how things cook, and many things you'd cook in the oven don't want that direct heat.
Broil mode should be fine with the stone in the oven. I wouldn't want to put a cold stone right up by the broiler as the thermal shock might crack it, but with it at the bottom of the oven there should be no issues, even if the stone isn't heated up.
Best Answer
Electric is generally better for baking. Gas ovens introduce a bit of water vapor due to the combustion of the gas. In general, electric ovens also have smaller variations in temperature during the oven cycle than gas, and so maintain a more even heat.