There are many different coordinate systems (X, Y, Z axes as you refer to them). The center of the Earth is often used as the origin "center" of the coordinate system, at least for calculations concerning the space in the vicinity of Earth. There are many such Earth-centered coordinate systems as well. In the Space Shuttle Program we used the "Mean of 50" and the "J2000" inertial systems often. They are well described in this article.

Although I have no personal experience with it, I imagine for planetary calculations, there is a Sun-centered coordinate system, and one for each planet that people care to calculate such things for.

To answer the last part of your question, such terminology as "Earth being North of Mars" is used only informally and would require some up-front definitions to be meaningful.

There are many different coordinate systems (X, Y, Z axes as you refer to them). The center of the Earth is often used as the origin "center" of the coordinate system, at least for calculations concerning the space in the vicinity of Earth. There are many such Earth-centered coordinate systems as well. In the Space Shuttle Program we used the "Mean of 50" and the "J2000" inertial systems often. They are well described in this article.

Although I have no personal experience with it, I imagine for planetary calculations, there are Sun-centered coordinate systems, and ones for each planet that people care to calculate such things for.

To answer the last part of your question, such terminology as "Earth being North of Mars" is used only informally and would require some up-front definitions to be meaningful.