Understanding the Mercator Projection
The Mercator Projection has been used and it is used since the XVI century due to it maintain the course, rhumb or route. So it is useful for navigation.

As the Earth is approached by a ellipsoid, then you get the formulas that you saw on Mercator Projection. This transformation it is complicated for a ellipsoid, as the Earth curvature it is not a constant, so in the example you gave (moving 300m eastwords, or 400m northwards), you need to transform the first point from geographical coordinates to rectangular coordinates, then shift the point 300 meters east, check the correction on distance due to the different curvature on each point, apply the correction, then shift the point 400 meters northwards, check the correction, apply the correction and bring the coordinates back to geographical coordinate.

That is to much computing time and there are a lost of precision loss on the different shiftings and transformation geographical &lt;-> rectangular coordinates. Web Mercator is just a Mercator that considers the Earth approach by an sphere, not a ellipsoid. That is much easier, needs less computation as, the curvature in a sphere is constant it is not required to consider. 

Mercator projection is ok for navigation (that is what you are doing with Google Earth or Google Maps). Check this link.

Regarding the constants... a is the semi-major axis of an ellipsoid, amap is the semi-major axis on the map (as the Mercator Projection is distorting the distances, you should make a distance correction, that it is become bigger as far as you are from the Equator) e is the eccentricity of the ellipsoid. 

Regarding the latitude issue... Thing a little bit on this question: In the Mercator Projection it is consider, as Earth approach a sphere and a revolution ellipsoid. Both surfaces are revolution surfaces so, there is no variation along the longitude way. That is why the longitude is not required for Y calculation.

Anyhow, is something is not clear, just let me know. Hope it helps!