This is popscience nonsense. Distances don't increase as you get closer to the massive body, they decrease. Besides, spacetime curvature relates to the tidal force. If "spacetime was more curved" at some location, the tidal force at that location would be stronger. It's really simple once it clicks. Like Einstein said, the speed of light varies with potential. You plot the gradient in the speed of light using light clocks placed throughout an equatorial slice of space. Then you get the rubber-sheet picture. The slope of the rubber sheet denotes the local force of gravity, the curvature of the rubber sheet denotes the tidal force.
Yes, the light is travelling slower. The path isn't quite straight though, it's slightly curved, but not so much that you can see it. Note that if you could, you wouldn't be able to distinguish any difference in curvature for the upper path and the lower path.
The "energy density" of space is higher. Because a concentration of energy in the middle alters the surrounding space.