I'm guessing Rasterization based pipelines use the same procedure for shading i.e. A ray is shot from the pixel and then from the triangle that got projected on that pixel to the light source?
Close, but not quite. There's no actual tracing of rays but the underlying linear algebra is the same. It sounds like you already understand the first part of rasterisation, as far as getting some fragments which need to be shaded. Each fragment knows the attributes of the triangle it came from (the world position, normal, etc.). In traditional forward shading, the fragment shader runs once per light, and it also knows the attributes of the light, including the world-space direction from the shading point to the light: the direction you would trace a shadow ray.
Although no shadow ray is traced, the actual lighting calculation is exactly the same: you use the view and light directions and the surface normal in whatever BRDF you want. You just repeat the process for each light and sum the results. At the end, you have the same result that you'd get from a very simple ray-tracer, without any shadows, reflections, refractions, or indirect lighting.
How to add those missing features? Well, there are several possible techniques for each, but I'll briefly go over the simplest. A simple way of getting shadows is using shadow maps. Before you render your main scene, you do a render into an off-screen buffer, but using the light as a camera. (The projection and shape depend on the light, e.g. orthogonal projection for a directional/cylindrical light.) The trick is, you don't render all the colours: you only draw into the depth buffer. What you get is a picture of the distance from the light to the closest thing in each direction. Later on, when you're shading for that light's contribution in the fragment shader, you calculate the distance from the light to the shading point, and compare it to the value in the depth buffer. That tells you whether the shading point is the closest thing to the light along that ray, without having to run an intersection test.
Planar reflections and refractions are pretty simple too: just render the scene but with the camera transformed to produce the reflection or refraction image. For example, for a planar reflection in a lake or floor, you reflect the camera position about that plane to draw the whole scene upside-down. When you draw the real scene around it, any transparency shows that reflection.
