PG,
I think MrAl provided a very good description and used a 2D description to help you. I think this is a good idea.
I see that my answer was not helpful, probably because I answered from the point of view of what charge and charge density are and not from the point of view of what the divergence theorem is telling you. .... My bad!
Once you have a firm grip on MrAl's examples, I can try to offer a way to see the answers in 3d using the spherically symmetric point charge Q at the origin.
For Q1, I hope you see that the divergence expression is indeterminate at the origin, you have 0/0 because the derivative goes to zero and the r^2 in the denominator (outside the derivative) goes to zero. If indeterminate, then you need to evaluate by a limiting process. Well, we know the answer already, so you actually don't need to go through this effort, but you can do it if you like.
For Q2, remember that the divergence theorem applies to all shapes, whether circles in 2D or spheres in 3D, or any other shape in 3D etc. However, you can make a useful shape to apply the divergence theorem over by considering two spheres of different radii, both with centers at the origin. Then take a particular solid angle and extend it from the origin to infinity. This solid angle will intersect the two spheres and you can make a useful enclosed volume with sides created by the cone created by the solid angle and the two spheres.
This enclosed volume is one you can apply the divergence theorem on in a very easy way. The value of D normal to and over the spherical sections is obvious and there is no normal component of D on the cone section of the surface, so the surface integral is zero there. Hence, the D vector enters into the volume on the inner sphere surface and the D vector exits the volume on the outer sphere surface. Do the calculations there and you will see that two surfaces have equal and opposite integrals.
You mentioned that the vector get smaller at larger radius, which is true. However, the area is getting bigger at the same rate.
Now, it may not be intuitively obvious that the divergence is zero for any surface that does not enclose the point charge, but this is the whole reason why we like the divergence theorem. It tell us a useful truth that might not have been obvious to us at first.