Use of pizza stone in gas oven
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:


It could block hot air vents from the flame into the oven chamber;
The oven floor is generally not designed to support weight and could
be damaged;
The pizza stone is liable to crack or even explode from thermal
shock due to rapid heating if it's in direct contact with the metal over the burner.


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.