I am beginner and I am trying to understand how blender works. My basic idea was to make hole inside object using Boolean Modifier. I put cube, apply boolean, then I put sphere as boolean object. So nothing happens, but when I clicked on EYE icon of sphere I got perfect result. Now I try to export this as .obj (which is my goal) and sphere is visible inside windows 3D object viewer. My goal is to only have cube with hole inside. What am I doing wrong?
You haven't selected Selection only when you export .obj-file.
Select your object. Select File -> Export -> Wavefront (.obj). Check Selection Only checkbox and press Export OBJ
To fix the topology you need to make sure that there are no overlapping edges. You can see your object wireframe by checking Wireframe checkbox under Object Properties panel from Viewport Display. You should check this to your object (The Cube).
From same panel change Display As to Bounds for your cutter (The Sphere). This way you can see if there are overlapping edges. Raise or lover the sphere so the edges are as far from each other as possible.
And also .obj export is not very good with quads so you need to make all your faces into triangles. Select your object and in Edit Mode press A to select all. Select Face and Triangulate Faces.