Yes, it is a picture of the entire universe when it was 400,000 years old taken today from the earth. But in the same way we take pictures with photo cameras, the object which the picture was taken of is 3D but the resulting picture itself is 2D. In the case of the CMB, we can think of the picture as follows: for each latitude and longitude on the earth, you point a camera straight up and record the CMB photons coming from that direction. Then, for each point on the surface of the earth (2D) you have a number - and that's the picture. These photons are coming from a very distant place in the universe and started traveling to us a very long time ago; and the energy of those photons is proportional to the amount of energy there was at that point in the universe when the photon started its trip towards the earth. Then that picture is telling us what the distribution of matter-energy was 400,000 years after the big bang.

You are perfectly right that the picture is like the internal surface of a sphere, and I've seen balloons with the CMB painted on it, which is probably the best representation of the picture. However, we like to have things on flat paper, and for that we need a projection from the surface of the sphere to a flat space. This is equivalent to the projections used to represent world maps on flat surfaces. I'm not sure what the particular projection used for CMB is.

Another interesting fact is that that picture is not the "actual" picture taken: it has been through two processes. In fact, originally it looks like this. This is due to the well-known doppler effect. We are moving with respect to the CMB photons, so the photons coming from the direction we're moving into seem to be more energetic than the photons coming from the opposite direction. This fact allows us to measure the speed we're moving through the CMB which happens to be about 600km/s.
After correcting for the doppler effect, what's left is this. In fact, the universe was extremely homogeneous 400,000 year after the big bang. However if one looks carefully it is possible to detect inhomogeneities in that picture, as small as 1 in 10^5. Those inhomogeneities is what actually is represented in the pictures as the one I showed in the previous post.