Is there a way to just apply a brush texture evenly across selected topology? In example, if I had a sphere, can I just have the texture apply across the selected sphere without having to click again and again to fill it in?
I think you might be able to achieve the result you're after with the type of Brush Mapping, in the brush texture section - Give "Tiled" a try. That will repeat the selected texture in the manner of a tileable texture.