Why the water acts like a mirror from the air? Why the ray of light what is in the air (n = 1) is reflected in the water (n = 1.33) surface, and acts like a mirror?
The Snell law say what the ray is refracted deflected closer to the normal. Then there will never be any reflection (total internal), then not can work like a mirror.
 A: As you quite rightly have written this is not an example of total internal reflection as the light is not attempting to go g=from an optically more dense medium to an optically less dense medium.
It is however an example of reflection from a boundary between air and water which will always occur.
In this case the reflection is probably more prominent because there is much less light coming from the bottom of the lake.  
Have you noticed that when standing inside a room at night you can often see your reflection in a pane of glass but during the day you cannot?
There is always a reflection when you stand inside but during the day the intensity of the light coming from outside is so much greater than the intensity of the reflected image that you do not see the reflection.
A: The critical angle is $\arcsin(n_2/n_1)$ where $n_2<n_1$ as you are going from the water to the air. This is the angel of 'total internal reflection'. On the other hand, the Brewster angle is $\arctan(n_2/n_1)$ where $n_2>n_1$ as you are going from the air to the water. This is the angle of perfect separation between polarizations. Light polarized perpendicular to the plane will be mostly reflected and light polarized in-plane will be completely refracted. When you combine these two concepts it means light from low angles beneath the water will never reach you (total internal reflection) and light from low angles above the water which enters the water will rarely reach you (subcritical polarization). That is why at low angles the light you see appears to be coming from above the surface. If you want an introductory mathematical answer, Wikipedia the 'Fresnel Equations'.
