Actually I do this, and I do it because my oven is old and tempermental.
Adding a heavy heat-sink (like a pizza stone, or a half dozen fire bricks) to your oven will increase your pre-heat time, but it makes your ovens temperature much more stable. It's a good thing to do if you're planning on cooking anything that is really temperature sensitive.
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
We had an older (1950's) gas oven for a decade which had very uneven heating and used a large pizza stone ( 13" by 16" ) to even out the heat. We did this by putting the stone on an oven rack in the lowest position, and making sure to give the oven at least 25min of heat-up time so the stone would be as hot as the oven. This did, indeed, help even out hot spots in the oven and decrease its tendency to burn delicate foods like cookies. It also eliminated the requirement to remember to take the pizza stone in and out of the oven.
You do not want to put the pizza stone directly on the oven floor. Three reasons:
Also, do be aware that your pizza stone will accumulate crud from food dripping/dropping in the oven, which burns into a black carbon powder. We turned the stone upside down every few weeks limit accumulation.