Is ozone evenly distributed throughout the atmosphere?

No. It is at its minimum at Earth's surface (where there is little energetic sunlight to make more, and plenty of things that cause it to decay), and in the exosphere (where there is so much energy almost everything is missing an electron or two, and collisions between atoms are too energetic to allow many captures). It has a maximum in the lower stratosphere, where UV-C shatters nitrogen and oxygen, and some of the oxygen later forms ozone.

Annually, each pole develops an "ozone hole" in late winter / early spring. Here the amount of ozone is reduced quite a bit by various sorts of contaminants... some sourced by Nature and some by Man.