Certainly this won't be a great place to live as the planets that did exist in that system would have very erratic orbits. Those planets would be swung close by and then far away from each of the two suns resulting in large changes in temperatures.
Maybe some form of virus or bacteria could evolve there, but anything approximating a complex life form would have to evolve under the surface.
How will an atmosphere form?
It all depends on the distance. If Jupiter were as close as mars the effects on earth would probably be enough to make earth a very different place.
The orbital mechanics would be pretty much the same if Jupiter were a small sun. When it comes to binary star systems we’re talking about even greater distances.
When we talk about orbiting the sun we really aren’t accurate. Earth, the sun and other planets actually orbit a common gravitational center.