Can stars be seen in the background when looking directly at our solar system?
Stars are much less bright than the earth in full sunlight as per your picture. So with the exposure required to give that picture of earth, no, you won't see stars.

But move the camera a bit to the side, so earth is not in shot and you can see stars.

The same goes for the sun - make sure sunlight is not entering the lens, and you can easily see stars.

Sunlight doesn't block any light out (unless you want to see a star that is actually behind the sun) it's just so bright that you need to make sure it isn't in shot.