You have a misconception as to what an ozone layer is. Ozone is a gas and is found throughout the atmosphere, even at ground level. The ozone layer is the range in the atmosphere where ozone levels are the highest. Oxygen O2 is present in the ozone layer as well (as are all other atmospheric gasses). You should also note that in the ozone layer there isn't actually a whole lot of ozone, just more than there is anywhere else. I think it's around 5 parts per million; meaning there is 1 molecule of ozone for every 200,000 other molecules. There isn't a blanket of ozone, it's just an area where ozone is 5-10ppm instead of 1ppm or less.

Why does most of it stay where it is? Ozone is a somewhat unstable molecule and the same mechanism that typically forms ozone also breaks it down. Simply, O2 gets hit by some photons of the right energy (some kind of UV light) and is split into O and O, one (or both) of those O's bonds to another O2 molecule to form O3, ozone. Sooner or later, that O3 gets hit by the right energy of UV light and one of the O's break off to form O2 and a spare O which will find another O2 to bond with and repeat the cycle. A small percent of the ozone does drop into the statosphere (where we live).

Because Oxygen we breath has a chemical formula of O2, and Ozone has a chemical formula of O3, and since three is bigger than for, than it must be above 2. Thus is why the ozone layer is above the "oxygen" layer.