It may be possible.
I am still unclear on the reasoning to use an arduino, ethernet shield, writing arduino code, etc.
Mach3 will drive 3 axis simultaneously, or one axis singly, in any direction (or Vector in the case of 2 or 3 axis), and rate you decide upon. CAD programs, some even free (Draftsight is a superb freebie) can produce dxf files from a line drawing, which can be converted into a G-code by a CAM program (I like CamBam), loaded and run.
There are topics in this forum describing how to package multiple programs and run them automatically in a "batch file" manner.
Much (maybe even most) of what is now done with Mach3 doesn't involve "hand writing" of G-code, except some specific scripts like running multiple programs as above.
John