How do Map Projections and Layer Projections relate

CRS (Coordinate Reference Systems) are a bit complex. At the most basic level, the Map Projection is the CRS in which the Map is displayed and is also the default output CRS for all analysis operations (as well as new layers).

Cartographica translates (internally) the CRS from each layer into the CRS of the Map before the layers are displayed and during analysis operations. However, there are some situations where the internal conversions are insufficient (such as some CRS that can't be translated well in both directions for all coordinates).

The key reasons why you might want to change your Map Projection are:

In many cases, changing the Map Projection will change the visual display of the layers, but shouldn't have an effect on how things line up, since Cartographica is translating the coordinates on-the-fly. If you do have coordinate shifts, it's often because one or more layers are Unprojected, meaning they have no designated CRS.

Hope this sheds some light.