This is a common misconception. In this analogy the fabric doesn't represent space-time, just reduced space (2D dimensions of it). It shows just the spatial curvature which has some effects on its own, but the main effect of gravitation (mass attraction), cannot be explained without the time dimension. Here are links to better visualizations:
Yes you have to imagine we live in an universe with less dimensions to understand this curvature visualizations by embedding. The illustrations are 3D, so a curved diagram in it can be 2D at max. The space-time diagrams linked above are not different that that aspect. They just use 1 space & 1 time dimension instead of 2 space dimensions.
If there was something above
the sun in that diagram, it would not be in our universe. The 3D dimensions of these illustrations are a virtual embedding space without physical meaning. It is needed to visualize curvature of the diagram.
This sun and earth are actually not spheres above the fabric, but circles within the fabric in a space-space diagram, even if they are often shown as spheres.
Well you don't. You can only visualize 2 of the 4 space-time dimensions in a 2D diagram.