Light path

Yes, that's the path that leads to constructive interference between neighbor paths. all other paths lead to destructive interference between neighbor paths.

 

For the path of minimum time (because it is a extremum) all neighbor paths take the same amount of time (to first order - zero derivative of time taken with respect to path - that's the definition of extremum). That means that they all have the same phase and interfere constructively. That's not the case for other paths which end up interfering destructively.

To understand why a specific amount of light gets reflected, you need to understand the boundary conditions for the electromagnetic fields. That's a fairly complicated topic. Look up Fresnel equations.

 

When light goes from one medium to another, there are two paths which lead to constructive interference - one which goes into the other medium, and the other is being reflected. To calculate how much is reflected, you need to know the refractive index of the materials. here is more detail: http://en.wikipedia.org/wiki/Fresnel_equations

edit: haha, dauto beat me to it :)