Your composition is set to use the C4D rendering engine. Change it to Classic 3D and the shadows will appear. They are not supported using the C4D rendering option. If you open the comp settings (Ctrl/Cmnd + K) you will see the list of unsupported features when using the C4D option.
I don't see anything wrong with your comp. I watched a couple of the chapters (lessons) in the video. I would give the tutorial about a C + because it leaves out a bunch of very important information and there seems to be very little in the way of explaining workflow or workspaces. The tutorial just leaves a little too much unsaid in that section.
Your shadows are dark because the falloff distance is big enough that the light is reaching the other layers, the diffusion is set to zero, and Cast Shadows is turned on with the text layers.
This is a lot easier to visualize if you turn on Multiple views so you can see the alignment of the lights, layers, and camera. You can quickly find all modified properties of a layer by pressing the U key twice. It is also a lot easier to set up your lighting first by setting the falloff to none, line things up, then try a few different falloff settings. I would also consider adding some ambient light and place a 3D solid down as a floor so you can see how the shadows fall.
You're not really looking at 2.5D animation either. To be accurate, 2.5D animation is the process of separating an image into foreground, middle ground and background elements then placing those elements in a 3D environment and moving a camera around.